Lie 2010
Lie 2010
Lie 2010
Contents
1 Groups 4
5 Commuting elements 22
7 Lie subgroups 28
9 Closed subgroups 37
14 Coset spaces 53
1
17 Normal subgroups and ideals 58
20 Representations 69
21 Schur orthogonality 77
22 Characters 81
26 Class functions 93
2
38.4 Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
38.5 Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3
1 Groups
The purpose of this section is to collect some basic facts about groups. We leave it to the reader
to prove the easy statements given in the text.
We recall that a group is a set G together with a map W G G ! G; .x; y/ 7! xy and an
element e D eG ; such that the following conditions are fulfilled
(a) .xy/z D x.yz/ for all x; y; z 2 GI
(b) xe D ex D x for all x 2 GI
1 1 1
(c) for every x 2 G there exists an element x 2 G such that xx Dx x D e:
Remark 1.1 Property (a) is called associativity of the group operation. The element e is called
the neutral element of the group.
The element x 1 is uniquely determined by the property (c); indeed, if x 2 G is given, and
y 2 G an element with xy D e; then x 1 .xy/ D x 1 e D x 1 ; hence x 1 D .x 1 x/y D ey D
y: The element x 1 is called the inverse of x:
Example 1.2 Let S be a set. Then Sym.S/; the set of bijections S ! S; equipped with
composition, is a group. The neutral element e equals IS ; the identity map S ! S; x 7! x:
If S D f1; : : : ; ng; then Sym.S/ equals Sn ; the group of permutations of n elements.
A group G is said to be commutative or abelian if xy D yx for all x; y 2 G: We recall that a
subgroup of G is a subset H G such that
(a) eG 2 H I
(b) xy 2 H for all x 2 H and y 2 H I
1
(c) x 2 H for every x 2 H:
We note that a subgroup is a group of its own right. If G; H are groups, then a homomorphism
from G to H is defined to be a map ' W G ! H such that
(a) '.eG / D eH I
(b) '.xy/ D '.x/'.y/ for all x; y 2 G:
We note that the image im.'/ WD '.G/ is a subgroup of H: The kernel of '; defined by
1
ker ' WD ' .feH g/ D fx 2 G j '.x/ D eH g
is also readily seen to be a subgroup of G: A surjective group homomorphism is called an epi-
morphism. An injective group homomorphism is called a monomorphism. We recall that a group
homomorphism ' W G ! H is injective if and only if its kernel is trivial, i.e., ker ' D feG g: A
bijective group homomorphism is called an isomorphism. The inverse ' 1 of an isomorphism
' W G ! H is a group homomorphism from H to G: Two groups G1 and G2 are called isomor-
phic if there exists an isomorphism from G1 onto G2 :
If G is a group, then by an automorphism of G we mean an isomorphism of G onto itself.
The collection of such automorphisms, denoted Aut.G/; is a subgroup of Sym.G/:
4
Example 1.3 If G is a group and x 2 G; then the map lx W G ! G; y 7! xy; is called left
translation by x: We leave it to the reader to verify that x 7! lx is a group homomorphism from
G to Sym.G/:
Likewise, if x 2 G; then rx W G ! G; y 7! yx; is called right translation by x: We leave it
to the reader to verify that x 7! .rx / 1 is a group homomorphism from G to Sym.G/:
If x 2 G; then Cx W G ! G; y 7! xyx 1 is called conjugation by x: We note that Cx is an
automorphism of G; with inverse Cx 1 : The map C W x ! Cx is a group homomorphism from G
into Aut.G/: Its kernel is the subgroup of G consisting of the elements x 2 G with the property
that xyx 1 D y for all y 2 G; or, equivalently, that xy D yx for all y 2 G: Thus, the kernel of
C equals the center Z.G/ of G:
We end this preparatory section with the isomorphism theorem for groups. To start with we recall
that a relation on a set S is a subset R of the Cartesian product S S: We agree to also write
xRy in stead of .x; y/ 2 R: A relation on S is called an equivalence relation if the following
conditions are fulfilled, for all x; y; z 2 S;
(a) x x (reflexivity);
(b) x y ) y x (symmetry);
(c) x y ^ y z ) x z (transitivity).
If x 2 S; then the collection x WD fy 2 S j y xg is called the equivalence class of x: The
collection of all equivalence classes is denoted by S= :
A partition of a set S is a collection P of non-empty subsets of S with the following proper-
ties
(a) if A; B 2 P; then A \ B D ; or A D BI
(b) [A2P A D S:
If is an equivalence relation on S then S= is a partition of S: Conversely, if P is a
partition of S; we may define a relation P as follows: x P y if and only if there exists a
set A 2 P such that x and y both belong to A: One readily verifies that P is an equivalence
relation; moreover, S= P D P:
Equivalence relations naturally occur in the context of maps. If f W S ! T is a map between
sets, then the relation on S defined by x y () f .x/ D f .y/ is an equivalence relation.
If x 2 S and f .x/ D c; then the class x equals the fiber
1 1
f .c/ WD f .fcg/ D fy 2 S j f .y/ D cg:
Let denote the natural map x 7! x from S onto S= : Then there exists a unique map
fN W S= ! T such that the following diagram commutes
f
S ! T
# % fN
S=
5
We say that f factors through a map fN W S= ! T: Note that fN.x/ D f .x/ for all x 2 S: The
map fN is injective, and has image equal to f .S/: Thus, if f is surjective, then fN is a bijection
from S= onto T:
Partitions, hence equivalence relations, naturally occur in the context of subgroups. If K is a
subgroup of a group G; then for every x 2 G we define the right coset of x by xK WD lx .K/:
The collection of these cosets, called the right coset space, is a partition of G and denoted by
G=K: The associated equivalence relation is given by x y () xK D yK; for all x; y 2 G:
The subgroup K is called a normal subgroup if xKx 1 D K; for every x 2 G: If K is a
normal subgroup then G=K carries a unique group structure for which the natural map W G !
G=K; x 7! xK is a homomorphism. Accordingly, xK yK D .x/.y/ D .xy/ D xyK:
Lemma 1.4 (The isomorphism theorem) Let f W G ! H be an epimorphism of groups. Then
K WD ker f is a normal subgroup of G: There exists a unique map fN W G=K ! H; such that
fN D f: The factor map fN is an isomorphism of groups.
Proof: Let x 2 G and k 2 K: Then f .xkx 1 / D f .x/f .k/f .x/ 1 D f .x/eH f .x/ 1 D eH ;
hence xkx 1 2 ker f D K: It follows that xKx 1 K: Similarly it follows that x 1 Kx K;
hence K xKx 1 and we see that xKx 1 D K: It follows that K is normal.
Let x 2 G and write f .x/ D h: Then, for every y 2 G; we have yK D xK () f .y/ D
f .x/ () y 2 f 1 .h/: Hence G=K consists of the fibers of f: In the above we saw that there
exists a unique map fN W G=K ! H; such that fN D f: The factor map is bijective, since f is
surjective. It remains to be checked that fN is a homomorphism. Now fN.eK/ D f .eG / D eH ;
since f is a homomorphism. Moreover, if x; y 2 G; then fN.xKyK/ D fN.xyK/ D f .xy/ D
f .x/f .y/: This completes the proof.
6
If G1 and G2 are Lie groups, we may equip the product manifold G D G1 G2 with the
product group structure, i.e., .x1 ; x2 /.y1 ; y2 / WD .x1 y1 ; x2 y2 /; and eG D .eG1 ; eG2 /:
Lemma 2.4 Let G1 ; G2 be Lie groups. Then G WD G1 G2 ; equipped with the above manifold
and group structure, is a Lie group.
Lemma 2.5 Let G be a Lie group, and let H G be both a subgroup and a smooth submani-
fold. Then H is a Lie group.
Example 2.6
(a) The unit circle T WD fz 2 C j jzj D 1g is a smooth submanifold as well as a subgroup of
the Lie group C : Therefore it is a Lie group.
(b) The q-dimensional torus Tq is a Lie group.
So far, all of our examples of Lie groups were commutative. We shall formulate a result that
asserts that interesting connected Lie groups are not to be found among the commutative ones.
For this we need the concept of isomorphic Lie groups.
Remark 2.8 (a) If ' W G ! H is a Lie group isomorphism, then ' is smooth and bijective and
its inverse is smooth as well. Hence, ' is a diffeomorphism.
(b) The collection of Lie group automorphisms of G; equipped with composition, forms a
group, denoted Aut.G/:
7
We recall that a topological space X is said to be connected if ; and X are the only subsets
of X that are both open and closed. The space X is said to be arcwise connected if for each pair
of points a; b 2 X there exists a continous curve c W 0; 1 ! X with initial point a and end
point b; i.e., c.0/ D a and c.1/ D b: If X is a manifold then X is connected if and only if X is
arcwise connected.
We can now formulate the promised results about connected commutative Lie groups.
Theorem 2.9 Let G be a connected commutative Lie group. Then there exist integers p; q 0
such that G is isomorphic to Tp Rq :
The proof of this theorem will be given at a later stage, when we have developed enough
technology. See Theorem 6.1.
A more interesting example is the following. In the sequel we will often discuss new general
concepts in the context of this important particular example.
Example 2.10 Let n be a positive integer, and let M.n; R/ be the set of real n n matrices.
Equipped with entry wise addition and scalar multiplication, M.n; R/ is a linear space, which in
2
an obvious way may be identified with Rn : For A 2 M.n; R/ we denote by Aij the entry of A
in the i -th row and the j -th column. The maps ij W A 7! Aij may be viewed as a system of
(linear) coordinate functions on M.n; R/:
In terms of these coordinate functions, the determinant function det W M.n; R/ ! R is given
by X
det D sgn ./1.1/ n.n/ ;
2Sn
where Sn denotes the group of permutations of f1; : : : ; ng; and where sgn denotes the sign of a
permutation. It follows from this formula that det is smooth.
The set GL.n; R/ of invertible matrices in M.n; R/; equipped with the multiplication of ma-
trices, is a group. As a set it is given by
Thus, GL.n; R/ is the pre-image of the open subset R D R n f0g of R under det: As the latter
function is continuous, it follows that GL.n; R/ is an open subset of M.n; R/: As such, it may
be viewed as a smooth manifold of dimension n2 : In terms of the coordinate functions ij ; the
multiplication map W GL.n; R/ GL.n; R/ ! GL.n; R/ is given by
n
X
kl ..A; B// D ki .A/il .B/:
iD1
8
map A 7! Aco is a polynomial, hence smooth map from M.n; R/ to itself. By Cramers rule the
inversion W GL.n; R/ ! GL.n; R/; A ! A 1 is given by
1
.A/ D .detA/ Aco :
Example 2.11 Let V be a real linear space of finite dimension n: Let v D .v1 ; : : : ; vn / be
an ordered basis of V: Then there is a unique linear isomorphism ev from Rn onto V; mapping
the j -th standard basis vector ej onto vj : If w is a second basis, then L WD ev 1 ew is a linear
isomorphism of Rn onto itself, hence a diffeomorphism. It follows that V has a unique structure
of smooth manifold such that the map ev is a diffeomorphism, for any choice of basis v:
We denote by End.V / the set of linear endomorphisms of V; i.e., linear maps of V into
itself. Equipped with pointwise addition and scalar multiplication, End.V / is a linear space. Let
v D .v1 ; : : : ; vn/ be an ordered basis of V: Given A 2 End.V /; we write mat.A/ D matv .A/
for the matrix of A with respect to v: The entries Aij of this matrix are determined by Avj D
P n
iD1 Aij vi ; for all 1 j n: As in Example 2.10 we denote by M.n; R/ the set of all real nn
matrices. Equipped with entry wise addition and scalar multiplication, M.n; R/ is a linear space.
Accordingly, mat is a linear isomorphism from End.V / onto M.n; R/: Via this map, composition
in End.V / corresponds with matrix multiplication in M.n; R/: More precisely,
for all A 2 End.V /: It follows that determinant and trace are independent of the choice of
basis. Hence, there exist unique maps det; tr W End.V / ! R such that detA D detmatA and
trA D trmatA for any choice of basis.
We denote by GL.V /; or also Aut.V /; the set of invertible elements of End.V /: Then GL.V /
is a group. Moreover, fix a basis of V; then the associated matrix map mat W End.V / ! M.n; R/
is a diffeomorphism, mapping GL.V / onto GL.n; R/: It follows that GL.V / is an open subset,
hence a submanifold of End.V /: Moreover, as mat restricts to a group isomorphism from GL.V /
onto GL.n; R/; it follows from the discussion in the previous example that GL.V / is a Lie group
and that mat is an isomorphism of Lie groups from GL.V / onto GL.n; R/:
9
Remark 2.12 In the above example we have distinguished between linear maps and their ma-
trices with respect to a basis. In the particular situation that V D Rn ; we shall often use the map
mat D mate ; defined relative to the standard basis e of Rn to identify the linear space End.Rn /
with M.n; R/ and to identify the Lie group GL.Rn / with GL.n; R/:
We shall now discuss an important criterion for a subgroup of a Lie group G to be a Lie
group. In particular this criterion will have useful applications for G D GL.V /: We start with a
result that illustrates the idea of homogeneity.
Let G be a Lie group. If x 2 G; then the left translation lx W G ! G; see Example 1.3, is
given by y 7! .x; y/; hence smooth. The map lx is bijective with inverse lx 1 ; which is also
smooth. Therefore, lx is a diffeomorphism from G onto itself. Likewise, the right multiplication
map rx W y 7! yx is a diffeomorphism from G onto itself. Thus, for every pair of points
a; b 2 G both lba 1 and ra 1 b are diffeomorphisms of G mapping a onto b: This allows us
to compare structures on G at different points. As a first application of this idea we have the
following.
Lemma 2.13 Let G be a Lie group and H a subgroup. Let h 2 H be a given point (in the
applications h D e will be most important). Then the following assertions are equivalent.
(a) H is a submanifold of G at the point hI
(b) H is a submanifold of G:
Proof: Obviously, (b) implies (a). Assume (a). Let n be the dimension of G and let m be the
dimension of H at h: Then m n: Moreover, there exists an open neighborhood U of h in
G and a diffeomorphism of U onto an open subset of Rn such that .h/ D 0 and such that
.U \ H / D .U / \ .Rm f0g/: Let k 2 H: Put a D kh 1 : Then la is a diffeomorphism of G
onto itself, mapping h onto k: We shall use this to show that H is a submanifold of dimension
m at the point k: Since a 2 H; the map la maps the subset H bijectively onto itself. The set
Uk WD la .U / is an open neighborhood of k in G: Moreover, k D la 1 is a diffeomorphism
of Uk onto the open subset .U / of Rn : Finally,
This shows that H is a submanifold of dimension m at the point k: Since k was an arbitrary point
of H; assertion (b) follows.
Example 2.14 Let V be a finite dimensional real linear space. We define the special linear
group
SL.V / WD fA 2 GL.V / j detA D 1g:
Note that det is a group homomorphism from GL.V / to R : Moreover, SL.V / is the kernel of
det: In particular, SL.V / is a subgroup of GL.V /: We will show that SL.V / is a submanifold of
GL.V / of codimension 1: By Lemma 2.13 it suffices to do this at the element I D IV :
10
Since G WD GL.V / is an open subset of the linear space End.V / its tangent space TI G may
be identified with End.V /: The determinant function is smooth from G to R hence its tangent
map is a linear map from End.V / to R: In Lemma 2.15 below we show that this tangent map is
the trace tr W End.V / ! R; A 7! tr.A/: Clearly tr is a surjective linear map. This implies that det
is submersive at I: By the submersion theorem, it follows that SL.V / is a smooth codimension 1
submanifold at I:
Lemma 2.15 The function det W GL.V / ! R has tangent map at I given by TI det D tr W
End.V / ! R; A 7! trA:
Proof: Put G D GL.V /: In the discussion in Example 2.14 we saw that TI G D End.V / and,
similarly, T1 R D R: Thus TI det is a linear map End.V / ! R: Let H 2 End.V /: Then by the
chain rule,
d
TI .det/.H / D det.I C tH /:
dt t D0
Fix a basis v1 ; : : : ; vn of V: We denote the matrix coefficients of a map A 2 End.V / with respect
to this basis by Aij ; for 1 i; j n: Using the definition of the determinant, we obtain
where R is polynomial in t and the matrix coefficients Hij : Differentiating this expression with
respect to t and substituting t D 0 we obtain
We shall now formulate a result that allows us to give many examples of Lie groups. The
complete proof of this result will be given at a later stage. Of course we will make sure not to
use the result in the development of the theory until then.
Theorem 2.16 Let G be a Lie group and let H be a subgroup of G: Then the following asser-
tions are equivalent.
(a) H is closed in the sense of topology.
(b) H is a submanifold.
Proof: For the moment we will only prove that (b) implies (a). Assume (b). Then there exists
an open neighborhood U of e in G such that U \ HN D U \ H: Let y 2 HN : Since ly is a
diffeomorphism from G onto itself, yU is an open neighborhood of y in G; hence yU \ H ;:
Select h 2 yU \ H: Then y 1 h 2 U: On the other hand, from y 2 HN ; h 2 H it follows that
y 1 h 2 HN : Hence, y 1 h 2 U \ HN D U \H; and we see that y 2 H: We conclude that HN H:
Therefore, H is closed.
11
By a closed subgroup of a Lie group G we mean a subgroup that is closed in the sense of
topology.
Corollary 2.17 Let G be a Lie group. Then every closed subgroup of G is a Lie group.
Corollary 2.18 Let ' W G ! H be a homomorphism of Lie groups. Then the kernel of ' is a
closed subgroup of G: In particular, ker ' is a Lie group.
Proof: Put K D ker ': Then K is a subgroup of G: Now ' is continuous and feH g is a closed
subset of H: Hence, K D ' 1 .feH g/ is a closed subset of G: Now apply Corollary 2.17.
Remark 2.19 We may apply the above corollary in Example 2.14 as follows. The map det W
GL.V / ! R is a Lie group homomorphism. Therefore, its kernel SL.V / is a Lie group.
Example 2.20 Let now V be a complex linear space of finite complex dimension n: Then by
End.V / we denote the complex linear space of complex linear maps from V to itself, and by
GL.V / the group of invertible maps. The discussion of Examples 2.10 and 2.11 goes through
with everywhere R replaced by C: In particular, the determinant det is a complex polynomial map
End.V / ! C; hence continuous. Since C D C n f0g is open in C; the set GL.V / D det 1 .C /
is open in End.V /: As in Example 2.11 we now see that GL.V / is a Lie group.
The map det W GL.V / ! C is a Lie group homomorphism. Hence, by Corollary 2.18 its
kernel, SL.V / WD fA 2 GL.V / j detA D 1g; is a Lie group.
Finally, let v D .v1 ; : : : ; vn/ be a basis of V (over C). Then the associated matrix map
mat D matv is a complex linear isomorphism from End.V / onto the space M.n; C/ of complex
n n matrices. It restricts to a Lie group isomorphism GL.V / ' GL.n; C/ and to a Lie group
isomorphism SL.V / ' SL.n; C/:
Another very useful application of Corollary 2.17 is the following. Let V be a finite dimen-
sional real linear space, and let W V V ! W be a bilinear map into a finite dimensional
real linear space W: For g 2 GL.V / we define the bilinear map g W V V ! W by
g .u; v/ D .g 1 u; g 1 v/: From g1 .g2 / D .g1 g2 / one readily deduces that the
stabilizer of in GL.V /;
GL.V / D fg 2 GL.V / j g D g
Lemma 2.21 The groups GL.V / and SL.V / are closed subgroups of GL.V /: In particular,
they are Lie groups.
12
Proof: Define Cu;v D fg 2 GL.V / j .g 1 u; g 1 v/ D .u; v/g; for u; v 2 V: Then GL.V /
is the intersection of the sets Cu;v ; for all u; v 2 V: Thus, to establish closedness of this group,
it suffices to show that each of the sets Cu;v is closed in GL.V /: For this, we consider the func-
tion f W GL.V / ! W given by f .g/ D .g 1 u; g 1 v/: Then f D .; /; hence f is
continuous. Since f.u; v/g is a closed subset of W; it follows that Cu;v D f 1 .f.u; v/g/ is
closed in GL.V /: This establishes that GL.V / is a closed subgroup of GL.V /: By application
of Corollary 2.17 it follows that GL.V / is a Lie group.
Since SL.V / is a closed subgroup of GL.V / as well, it follows that SL.V / D SL.V / \
GL.V / is a closed subgroup, hence a Lie group.
By application of the above to particular bilinear forms, we obtain interesting Lie groups.
Example 2.22 (a) Take V D Rn and the standard inner product on Rn : Then GL.V / D
O.n/; the orthogonal group. Moreover, SL.V / D SO.n/; the special orthogonal group.
Example 2.23 Let n D p C q; with p; q positive integers and put V D Rn : Let be the
standard inner product of signature .p; q/; i.e.,
p n
X X
.x; y/ D xi yi xi yi :
iD1 iDpC1
Then GL.V / D O.p; q/ and SL.V / D SO.p; q/: In particular, we see that the Lorentz group
O.3; 1/ is a Lie group.
Example 2.24 Let V D R2n and let be the standard symplectic form given by
n
X n
X
.x; y/ D xi ynCi xnCi yi :
iD1 iD1
Example 2.25 Let V be a finite dimensional complex linear space, equipped with a complex
inner product : This inner product is not a complex bilinear form, since it is skew linear in its
second component (this will always be our convention with complex inner products). However,
as a map V V ! C it is bilinear over RI in particular, it is continuous. As in the proof of
Lemma 2.21 we infer that the associated unitary group U.V / D GL.V / is a closed subgroup
of GL.V /; hence a Lie group. Likewise, the special unitary group SU.V / WD U.V / \ SL.V / is
a Lie group.
Via the standard basis of Cn we identify End.Cn / ' M.n; C/ and GL.Cn / ' GL.n; C/; see
also Remark 2.12. We equip Cn with the standard inner product given by
n
X
hz ; wi D zi wN i .z; w 2 Cn /:
iD1
The associated unitary group U.Cn / may be identified with the group U.n/ of unitary n n-
matrices. Similarly, SU.Cn / corresponds with the special unitary matrix group SU.n/:
13
Remark 2.26 It is possible to immediately apply Lemma 2.21 in the above example, in order
to conclude that U.n/ is closed. For this we observe that we may forget the complex structure of
V and view it as a real linear space. We write V.R/ for V viewed as a linear space. If n D dimC V
and if v1 ; : : : ; vn is a basis of V; then v1 ; iv1 ; : : : ; vn; ivn is a basis of the real linear space V.R/ :
In particular we see that dimR V.R/ D 2n: Any complex linear map T 2 End.V / may be viewed
as a real linear map from V to itself, hence as an element of End.V.R/ /; which we denote by
T.R/ : We note that T 7! T.R/ is a real linear embedding of End.V / into End.V.R/ /: Accordingly
we may view End.V / as a real linear subspace of End.V.R/ /: Let J denote multiplication by i;
viewed as a real linear endomorphism of V.R/ : We leave it to the reader to verify that
Accordingly,
GL.V / D fa 2 GL.V.R/ / j a J D J Ag:
From this one readily deduces that GL.V / is a closed subgroup of GL.V.R/ /: In the situation
of Example 2.25, H WD GL.V.R/ / is a closed subgroup of GL.V.R/ /; by Lemma 2.21. Hence
U.V / D GL.V / \ H is a closed subgroup as well.
We end this section with useful descriptions of the orthogonal, unitary and symplectic groups.
Example 2.27 For a matrix A 2 M.n; R/ we define its transpose At 2 M.n; R/ by .At /ij D
Aj i : Let D h ; i be the standard inner product on Rn : Then hAx ; yi D hx ; At yi: Let
a 2 GL.n; R/: Then for all x; y 2 Rn ;
1
a .x; y/ D hax ; ayi D hat ax ; yi:
O.n/ D fa 2 GL.n; R/ j at a D I g:
Example 2.28 If A 2 M.n; C/ we denote its complex adjoint by .A /ij D ANj i : Let h ; i be
the complex standard inner product on Cn : Then hAx ; yi D hx ; A yi for all x; y 2 Cn : As in
the previous example we now deduce that
U.n/ D fa 2 GL.n; C/ j a a D I g:
Example 2.29 Let be the standard symplectic form on R2n ; see Example 2.24. Let J 2
M.2n; R/ be defined by
0 I
J D ;
I 0
where the indicated blocks are of size n n:
Let h ; i denote the standard inner product on R2n : Then for all x; y 2 R2n ; we have
.x; y/ D hx ; Jyi: Let a 2 GL.n; R/; then
1
a .x; y/ D hax ; Jayi D hx ; at Jayi:
14
From this we see that Sp.n; R/ D GL.2n; R/ consists of all a 2 GL.2n; R/ with at Ja D J;
or, equivalently, with
.at / 1 D JaJ 1 (1)
This description motivates the following definition. The map A 7! At uniquely extends to
a complex linear endomorphism of M.2n; C/: This extension is given by the usual formula
.At /ij D Aj i : We now define Sp.n; C/ to be the collection of a 2 GL.2n; C/ satisfying condi-
tion (1). One readily verifies that Sp.n; C/ is a closed subgroup of GL.2n; C/ hence a Lie group.
We call it the complex symplectic group.
Note that GL.2n; R/ is a closed subgroup of GL.2n; C/ and that Sp.n; R/ D GL.2n; R/ \
Sp.n; C/:
Finally, we define the compact symplectic group by
Sp.n/ WD U.2n/ \ Sp.n; C/:
Clearly, this is a closed subgroup of GL.2n; C/; hence a Lie group.
Remark 2.30 In this section we have frequently used the following principle. If G is a Lie
group, and if H; K G are closed subgroups, then H \ K is a closed subgroup, hence a Lie
group.
15
If X 2 Te G; we define X to be the maximal integral curve of vX with initial point e:
Lemma 3.2 Let X 2 Te G: Then the integral curve X has domain R: Moreover, we have
X .s C t / D X .s/X .t / for all s; t 2 R: Finally the map .t; X/ 7! X .t /; R Te G ! G is
smooth.
Proof: Let be any integral curve for vX ; let y 2 G; and put 1 .t / D y.t /: Differentiating
this relation with respect to t we obtain:
d d
1 .t / D T.t / ly .t / D T.t /ly vX ..t // D vX .1 .t //;
dt dt
by left invariance of vX : Hence 1 is an integral curve for vX as well.
Let now I be the domain of X ; fix t1 2 I; and put x1 D X .t1 /: Then 1 .t / WD x1 X .t /
is an integral curve for vX with starting point x1 and domain I: On the other hand, the maximal
integral curve for vX with starting point x1 is given by 2 W t 7! X .t C t1 /: It has domain I t1 :
We infer that I I t1 : It follows that s C t1 2 I for all s; t1 2 I: Hence, I D R:
Fix s 2 R; then by what we saw above c W t 7! X .s/X .t / is the maximal integral curve for
vX with initial pont X .s/: On the other hand, the same holds for d W t 7! X .s C t /: Hence, by
uniqueness of the maximal integral curve, c D d:
The final assertion is a consequence of the fact that the vector field vX depends linearly, hence
smoothly on the parameter X: Let 'X denote the flow of vX : Then it is a well known (local)
result that the map .X; t; x/ 7! 'X .t; x/ is smooth. In particular, .t; X/ 7! X .t / D 'X .t; e/ is
a smooth map R Te G ! G:
Definition 3.3 Let G be a Lie group. The exponential map exp D expG W Te G ! G is defined
by
exp.X/ D X .1/
where X is defined as above; i.e., X is the maximal integral curve with initial point e of the
left invariant vector field vX on G determined by vX .e/ D X:
Example 3.4 We return to the example of the group GL.V /; with V a finite dimensional real
linear space. Its neutral element e equals I D IV : Since GL.V / is open in End.V /; we have
Te GL.V / D End.V /: If x 2 GL.V /; then lx is the restriction of the linear map Lx W A 7!
xA; End.V / ! End.V /; to GL.V /; hence Te .lx / D Lx ; and we see that for X 2 End.V /
the invariant vectorfield vX is given by vX .x/ D xX: Hence, the integral curve X satisfies the
equation:
d
.t / D .t /X:
dt
Since t 7! e tX is a solution to this equation with the same initial value, we must have that
X .t / D e tX : Thus in this case exp is the ordinary exponential map X 7! e X ; End.V / !
GL.V /:
16
Remark 3.5 In the above example we have used the exponential e A of an endomorphism A 2
End.V /: One way to define this exponential is precisely by the method of differential equations
just described. Another way is to introduce it by its power series
X1
A 1 n
e D A :
nD0
n
From the theory of power series it follows that A ! e A is a smooth map End.V / ! End.V /:
Moreover,
d tA
e D Ae tA D e tA A;
dt
by termwise differentiation of power series. By multiplication of power series we obtain
Applying this with X D sA and Y D tA; we obtain e .sCt /A D e sA e tA ; for all A 2 End.V / and
s; t 2 R: This formula will be established in general in Lemma 3.6 (b) below.
Moreover, the map exp W Te G ! G is smooth and a local diffeomorphism at 0: Its tangent map
at the origin is given by T0 exp D ITe G :
d
c.t / D s P X .st / D s vX .X .st // D vsX .c.t //:
dt
Hence c is the maximal integral curve of vsX with initial point e; and we conclude that c.t / D
sX .t /: Now evaluate at t D 1 to obtain the equality.
Formula (b) is an immediate consequence of (a) and Lemma 3.2. Finally, from Lemma 3.2
we have that .t; X/ 7! X .t / is a smooth map R Te G ! G: Substituting t D 1 we obtain
smoothness of exp : Moreover,
d
T0 .exp/X D exp.tX/jt D0 D P X .0/ D vX .e/ D X:
dt
Hence T0 .exp/ D ITe X ; and from the inverse function theorem it follows that exp is a local
diffeomorphism at 0; i.e., there exists an open neighborhood U of 0 in Te G such that exp maps
U diffeomorphically onto an open neighborhood of e in G:
17
Definition 3.7 A smooth group homomorphism W .R; C/ ! G is called a one-parameter
subgroup of G:
Proof: The first assertion follows from Lemma 3.2. Let W R ! G be a one-parameter
subgroup. Then .0/ D e; and
d d d
.t / D .t C s/jsD0 D .t /.s/jsD0 D Te .l.t / /.0/
P D vX ..t //;
dt ds ds
hence is an integral curve for vX with initial point e: Hence D X by the uniqueness of
integral curves. Now apply Lemma 3.6.
We now come to a very important application.
Lemma 3.9 Let ' W G ! H be a homomorphism of Lie groups. Then the following diagram
commutes:
'
G ! H
expG " " expH
Te '
Te G ! Te H
Proof: Let X 2 Te G: Then .t / D '.expG .tX// is a one-parameter subgroup of H: Differen-
tiating at t D 0 we obtain .0/
P D Te .'/T0 .expG /X D Te .'/X: Now apply the above lemma to
conclude that .t / D expH .t Te .'/X/: The result follows by specializing to t D 1:
Example 4.2 We return to the example of GL.V /; with V a finite dimensional real linear
space. Since GL.V / is an open subset of the linear space End.V / we may identify its tan-
gent space at I with End.V /: If x 2 GL.V /; then Cx is the restriction of the linear map
Cx W A 7! xAx 1 ; End.V / ! End.V /: Hence Ad.x/ D Te .Cx / D Cx is conjugation by
x:
18
The above example suggests that Ad.x/ should be looked at as an action of x on Te G by
conjugation. The following result is consistent with this point of view.
Lemma 4.3 Let x 2 G; then for every X 2 Te G we have
1
x exp X x D exp.Ad.x/ X/:
Proof: We note that Cx W G ! G is a Lie group homomorphism. Hence we may apply
Lemma 3.9 with H D G and ' D Cx : Since Te Cx D Ad.x/; we see that the following diagram
commutes:
Cx
G ! G
exp " " exp
Ad.x/
Te G ! Te G
The result follows.
19
Example 4.7 Let V be finite dimensional real linear space. Then for x 2 GL.V / the linear
map Ad.x/ W End.V / ! End.V / is given by Ad.x/Y D xY x 1 : Substituting x D e tX and
differentiating the resulting expression with respect to t at t D 0 we obtain:
d tX
. adX/Y D e Ye tX t D0 D XY YX:
dt
Hence in this case . adX/Y is the commutator bracket of X and Y .
Motivated by the above example we introduce the following notation.
Definition 4.8 For X; Y 2 Te G we define the Lie bracket X; Y 2 Te G by
X; Y WD . adX/Y
Lemma 4.9 The map .X; Y / 7! X; Y is bilinear Te G Te G ! Te G: Moreover, it is anti-
symmetric, i.e.,
X; Y D Y; X .X; Y 2 Te G/:
Proof: The bilinearity is an immediate consequence of the fact that ad W Te G ! End.Te G/ is
linear. Let Z 2 Te G: Then for all s; t 2 R we have
exp.t Z/ D exp.sZ/ exp.t Z/ exp. sZ/ D exp.t Ad.exp.sZ// Z/;
by Lemmas 3.6 and 4.3. Differentiating this relation with respect to t at t D 0 we obtain:
Z D Ad.exp.sZ// Z .s 2 R/:
Differentiating this with respect to s at s D 0 we obtain:
0 D ad.Z/T0 .exp/Z D ad.Z/Z D Z; Z:
Now substitute Z D X C Y and use the bilinarity to arrive at the desired conclusion.
20
Corollary 4.11 For all X; Y; Z 2 Te G;
X; Y ; Z D X; Y; Z Y; X; Z: (6)
Proof: Put ' D Ad and H D GL.Te G/: Then eH D I and TI H D End.Te G/: Moreover,
A; BH D AB BA for all A; B 2 End.Te G/: Applying Lemma 4.10 and using that ; G D
; and Te ' D ad; we obtain
ad.X; Y / D adX; adY H D adX adY adY adX:
Applying the latter relation to Z 2 Te G; we obtain (6).
Definition 4.12 A real Lie algebra is a real linear space a equipped with a bilinear map ; W
a a ! a; such that for all X; Y; Z 2 a we have:
(a) X; Y D Y; X (anti-symmetry);
(b) X; Y; Z C Y; Z; X C Z; X; Y D 0 (Jacobi identity).
Remark 4.13 Note that condition (a) may be replaced by the equivalent condition (a): X; X D
0 for all X 2 a: In view of the anti-symmetry (a), condition (b) may be replaced by the equiva-
lent condition (6). We leave it to the reader to check that another equivalent form of the Jacobi
identity is given by the Leibniz type rule
X; Y; Z D X; Y ; Z C Y; X; Z: (7)
Corollary 4.14 Let G be a Lie group. Then Te G equipped with the bilinear map .X; Y / 7!
X; Y WD . adX/Y is a Lie algebra.
Proof: The anti-linearity was established in Lemma 4.9. The Jacobi identity follows from (6)
combined with the anti-linearity.
Definition 4.15 Let a; b be Lie algebras. A Lie algebra homomorphism from a to b is a linear
map ' W a ! b such that
'.X; Y a / D '.X/; '.Y /b ;
for all X; Y 2 a:
From now on we will adopt the convention that Roman capitals denote Lie groups. The
corresponding Gothic lower case letters will denote the associated Lie algebras. If ' W G ! H
is a Lie group homomorphism then the associated tangent map Te ' will be denoted by ' : We
now have the following.
Lemma 4.16 Let ' W G ! H be a homomorphism of Lie groups. Then the associated tan-
gent map ' W g ! h is a homomorphism of Lie algebras. Moreover, the following diagram
commutes:
'
G ! H
expG " " expH
'
g ! h
Proof: The first assertion follows from Lemma 4.10, the second from Lemma 3.9.
21
Example 4.17 We consider the Lie group G D Rn : Its Lie algebra g D T0 Rn may be identified
with Rn : From the fact that G is commutative, it follows that Cx D IG ; for all x 2 G: Hence,
Ad.x/ D Ig ; for all x 2 G: It follows that ad.X/ D 0 for all X 2 g: Hence X; Y D 0 for all
X; Y 2 g:
Let X 2 g ' Rn : Then the associated one-parameter subgroup X is given by X .t / D tX:
Hence exp.X/ D X; for all X 2 g:
We consider the Lie group homomorphism ' D .'1 ; : : : ; 'n / W Rn ! Tn given by 'j .x/ D
e 2 ixj : One readily verifies that ' is a local diffeomorphism. Its kernel equals Zn : Hence, by
the isomorphism theorem for groups, the map ' factors through an isomorphism of groups 'N W
Rn =Zn ! Tn : Via this isomorphism we transfer the manifold structure of Tn to a manifold
structure on Rn =Zn : Thus, Rn =Zn becomes a Lie group, and 'N an isomorphism of Lie groups.
Note that the manifold structure on H WD Rn =Zn is the unique manifold structure for which the
canonical projection W Rn ! Rn =Zn is a local diffeomorphism. The projection is a Lie
group homomorphism. The associated homomorphism of Lie algebras W g ! h is bijective,
since is a local diffeomorphism. Hence, is an isomorphism of Lie algebras. We adopt
the convention to identify h with g ' Rn via : It then follows from Lemma 4.16 that the
exponential map expH W Rn ! H D Rn =Zn is given by expH .X/ D .X/ D X C Zn :
5 Commuting elements
In the following we assume that G is a Lie group with Lie algebra g: Two elements X; Y 2 g are
said to commute if X; Y D 0: The Lie algebra g is called commutative if every pair of elements
X; Y 2 g commutes.
Example 5.1 If G D GL.V /; with V a finite dimensional real or complex linear space, then
g D End.V /: In this case the Lie bracket of two elements A; B 2 End.V / equals the commutator
bracket A; B D AB BA: Hence, the assertion that A and B commute means that AB D BA;
as we are used to. In this case we know that the associated exponentials e A and e B commute as
linear maps, hence as elements of GI moreover, e A e B D e ACB : The following lemma generalizes
this fact to arbitrary Lie algebras.
Lemma 5.2 Let X; Y 2 g be commuting elements. Then the elements exp X and exp Y of G
commute. Moreover,
exp.X C Y / D exp X exp Y:
Proof: We will first show that x D exp X and y D exp Y commute. For this we observe that, by
Lemma 4.3, xyx 1 D exp.Ad.x/Y /: Now Ad.x/Y D e adX Y; by Lemma 4.6. Since ad.X/Y D
X; Y D 0; it follows that ad.X/n Y D 0 for all n 1: Hence, Ad.x/Y D e adX Y D Y:
Therefore, xyx 1 D y and we see that x and y commute.
For every s; t 2 R we have that sX; t Y D st X; Y D 0: Hence by the first part of this proof
the elements exp.sX/ and exp.t Y / commute for all s; t 2 R: Define the map W R ! G by
22
Then .0/ D e: Moreover, for s; t 2 R we have
.s C t / D exp.s C t /X exp.s C t /Y
D exp sX exp tX exp sY exp t Y
D exp sX exp sY exp tX exp t Y D .s/.t /:
It follows that is a one-parameter subgroup of G: Hence D Z with Z D 0 .0/; by Lemma
3.8. Now, by Lemma 5.3 below,
0 d d
.0/ D exp.tX/ exp.0/ C exp.0/ exp.t Y / D X C Y:
dt t D0 dt t D0
From this it follows that .t / D Z .t / D exp.t Z/ D exp.t .X C Y //; for t 2 R: The desired
equality follows by substituting t D 1:
The following lemma gives a form of the chain rule for differentiation that has been used in
the above, and will often be useful to us.
Definition 5.4 The subgroup Ge generated by the elements exp X; for X 2 g; is called the
component of the identity of G:
23
By an open subgroup of a Lie group G we mean a subgroup H of G that is an open subset
of G in the sense of topology.
Proof: Let a 2 Ge : Then there exists a positive integer k 1 and elements X1 ; : : : ; Xk 2 g such
that a D exp.X1 / : : : exp.Xk /: The map exp W g ! G is a local diffeomorphism at 0 hence there
exists an open neighborhood of 0 in g such that exp is a diffeomorphism of onto an open
subset of G: Since la is a diffeomorphism, it follows that la .exp.// is an open neighborhood of
a: We now observe that la .exp.// D fexp.X1 / : : : exp.Xk / exp.X/ j X 2 g Ge : Hence a
is an interior point of Ge : It follows that Ge is open in G:
Proof: The set Ge is open and closed in G; hence a (disjoint) union of connected compo-
nents. On the other hand Ge is arcwise connected. For let a 2 Ge ; then we may write
a D exp.X1 / : : : exp.Xk / with k 1 and X1 ; : : : ; Xk 2 g: It follows that c W 0; 1 ! G; t 7!
exp.tX1 / : : : exp.tXk / is a continuous curve with initial point c.0/ D e and end point c.1/ D a:
This establishes that Ge is arcwise connected, hence connected. Therefore Ge is a connected
component; it obviously contains e:
Lemma 5.9 Let G be a Lie group, x 2 G: Then the following assertions are equivalent.
(a) x commutes with Ge I
(b) Ad.x/ D I:
Proof: Assume (a). Then for every Y 2 g and t 2 R we have exp.t Y / 2 Ge ; hence
1
exp.t Ad.x/Y / D x exp t Y x D exp t Y
Differentiating this expression at t D 0 we see that Ad.x/Y D Y: This holds for any Y 2 g;
hence (b).
For the converse implication, assume (b). If Y 2 g; then
1
x exp Y x D exp Ad.x/Y D exp Y:
Hence x commutes with exp.g/: Since the latter set generates the subgroup Ge ; it follows that x
commutes with Ge :
24
Remark 5.10 Note that the point of the above proof is that one does not need exp W g ! G
to be surjective in order to derive properties of a connected Lie group G from properties of its
Lie algebra. It is often enough that G is generated by exp g: Another instance of this principle is
given by the following theorem.
Theorem 5.11 Let G be a Lie group. The following conditions are equivalent.
(a) The Lie algebra g is commutative.
(b) The group Ge is commutative.
In particular, if G is connected then g is commutative if and only if G is commutative.
Proof: Assume (a). Then X; Y D 0 for all X; Y 2 g: Hence exp X and exp Y commute for
all X; Y 2 g and it follows that Ge is commutative.
Conversely, assume (b). Let x 2 Ge : Then it follows by the previous lemma that Ad.x/ D I:
In particular this holds for x D exp.tX/; with X 2 g and t 2 R: It follows that e ad.tX/ D
Ad.exp.tX// D I: Differentiating at t D 0 we obtain ad.X/ D 0: Hence X; Y D 0 for all
X; Y 2 g and (a) follows.
Finally, if G is connected, then Ge D G and the last assertion follows.
25
Proof: (a) ) (b): Let h 2 H: Then Uh D hU is an open neighborhood of h in G: Moreover,
Uh \ H D hU \ H D h.U \ h 1 H / D h.U \ H / D fhg:
(b) ) (c): We first prove that H is closed in G: Let U be an open neighborhood of e in
G such that U \ H D feg: Let g 2 G be a point in the closure of H: Then it suffices to show
that g 2 H: There exists a sequence fhj g in H converging to g: It follows that hj C1 hj 1 !
gg 1 D e; as j ! 1: Hence for j sufficiently large we have hj C1 hj 1 2 U \ H D feg; hence
hj D hj C1 : It follows that the sequence hj becomes stationary after a certain index; hence
hj D g for j sufficiently large and we conclude that g 2 H:
It follows from the above that the set H \ C is closed in C; hence compact. For h 2 H \ C
we select an open subset of Uh of G such that Uh \ H D fhg: Then fUh j h 2 H \ C g is an
open cover of H \ C which does not contain a proper subcover. By compactness of H \ C this
cover must therefore be finite, and we conclude that H \ C is finite.
(c) ) (d) Let g 2 G be a point in the closure of H: The point g has a compact neighborhood
C: Now g lies in the closure of H \ C I the latter set is finite, hence closed. Hence g 2 H \ C
H and we conclude that the closure of H is contained in H: Therefore, H is closed.
It follows that H is a closed Lie subgroup. Its Lie algebra h consists of the X 2 g with
exp.RX/ H: Since exp W g ! G is a local diffeomorphism at 0, there exists an open neigh-
borhood of 0 if g such that exp is injective on : Let X 2 g n f0g: Then there exists an > 0
such that ; X : The curve c W ; ! G; t 7! exp tX has compact image; this
image has a finite intersection with H: Hence ft 2 ; j exp tX 2 H g is finite, and we see
that X h: It follows that h D f0g:
(d) ) (a) Assume (d). Then H is a closed smooth submanifold of X of dimension 0: By
definition this implies that there exists an open neighborhood U of e in G such that U \H D feg:
Hence (a).
Proof of Theorem 6.1: Assume that G is a connected Lie group that is commutative. Then
its Lie algebra g is commutative, i.e., X; Y D 0 for all X; Y 2 g: From this it follows that
exp X exp Y D exp.X C Y / for all X; Y 2 g: Therefore, the map exp W g ! G is a homomor-
phism of the Lie group .g; C; 0/ to G: It follows that exp.g/ is already a subgroup of G; hence
equals the subgroup Ge generated by it. Since G is connected, Ge D G; and it follows that
exp has image G; hence is a surjective Lie group homomorphism. Let be the closed subgroup
ker.exp/ of g: By the isomorphism theorem for groups we have G ' g= as groups.
Since exp is a local diffeomorphism at 0; there exists an open neighborhood of 0 in g on
which exp is injective. In particular this implies that \ D f0g: By Proposition 6.2 it follows
that is a discrete subgroup of g: In view of Lemma 6.4 below there exists a collection
1 ; : : : ;
p
of linear independent elements in g such that D Z
1 Z
p : We may extend the above
set to a basis
1 ; : : : ;
n of gI here n D dimg D p C q for some q 2 N: Via the basis
1 ; : : : ;
n
we obtain a linear isomorphism ' W g ! Rp Rq : Let D exp ' 1 ; then W Rn ! G is
a surjective Lie group homomorphism, and a local diffeomorphism everywhere. Moreover, its
kernel equals './ D Zp f0g: It follows that factors through a bijective group homomorphism
N W .R=Z/p Rq ' Rn =.Zp f0g/ ! G: The canonical map W Rn ! .R=Z/p Rq is a
local diffeomorphism. Moreover, D N is a local diffeomorphism as well. Hence N is a local
diffeomorphism. Since N is a bijective as well, we conclude that N is a diffeomorphism, hence an
26
isomorphism of Lie groups.
Lemma 6.3 Let ' W G ! H be a homomorphism of Lie groups. If ' is a local diffeomorphism
at e; then ' is a local diffeomorphism at every point of G:
Proof: We prove this by homogeneity. Let a 2 G: Then from '.ax/ D '.a/'.x/ we see that
' l D l'.a/ '; hence ' D l'.a/ ' la 1 : Now la and l'.a/ are diffeomorphisms. Since la 1
maps a to e; whereas ' is a local diffeomorphism at e it follows that l'.a/ ' la 1 is a local
diffeomorphism at aI hence ' is a local diffeomorphism at a:
Lemma 6.4 Let V be a finite dimensional real linear space. Let be a discrete subgroup
of V: Then there exists a collection of linearly independent elements
1 ; : : : ;
p of V such that
D Z
1 Z
p :
the latter set is finite by compactness of C C 0; 1
1 : Thus we see that ./ \ C is finite for
every compact subset of W: By Prop. 6.2 this implies that ./ is a discrete subgroup of W: By
the induction hypothesis there exist linearly independent elements
N2 ; : : : ;
Np of ./ such that
./ D Z
N2 Z
Np : Fix
2 ; : : : ;
p 2 such that .
j / D
Nj : Then the elements
1 ; : : : ;
p
are readily seen to be linear independent; moreover, D Z
1 Z
p :
27
7 Lie subgroups
Definition 7.1 A Lie subgroup of a Lie group G is a subgroup H; equipped with the structure
of a Lie group, such that the inclusion map W H ! G is a Lie group homomorphism.
The above definition allows examples of Lie subgroups that are not submanifolds. This is
already so if we restrict ourselves to one-parameter subgroups.
Lemma 7.2 Let G be a Lie group, and let X 2 g: The image of the one-parameter subgroup
X is a Lie subgroup of G:
Proof: The result is trivial for X D 0: Thus, assume that X 0: The map X W R ! G is a Lie
group homorphism. Its image H is a subgroup of G:
Assume first that X is injective. Then H has a unique structure of smooth manifold for
which the bijection X W R ! H is a diffeomorphism. Clearly, this structure turns H into a Lie
group and the inclusion map i W H ! G is a Lie group homomorphism.
0
Next, assume that X is not injective. As X .0/ D X 0; the map t 7! X .t / D exp tX is
injective on a suitable open interval I containing 0: It follows that ker X is a discrete subgroup
of R: Hence ker X D Z
for some
2 R: This implies that there exists a unique group
homomorphism N W R=Z
! R such that X D N pr: Since pr is a local diffeomorphism, the
map N is smooth, hence a Lie group homomorphism. Therefore, H D im N is compact. By
homogeneity, N is an injective immersion. This implies that N is an embedding of R=Z
onto a
smooth submanifold of G: We conclude that H D im N is a smooth submanifold of G; hence a
Lie subgroup.
We will give an example of a one-parameter subgroup of T2 whose image is everywhere
dense. The following lemma is needed as a preliminary.
28
Proof: Let H denote the image of : For j D 1; 2; let pj W T2 ! T denote the projection onto
the j -th component and consider the one-parameter subgroup j WD pj W R ! T: Its kernel
j is an additive subgroup of R; hence either trivial or infinite. If j0 .0/ D 0 then j D R:
If j0 .0/ 0 then j is immersive at 0; hence everywhere by homogeneity. It follows that the
image of j is an open subgroup of T; hence equal to T; by connectedness of the latter group.
It follows that j is a local diffeomorphism from R onto T: On the other hand, j cannot be a
diffeomorphism, as T is compact and R is not. It follows that j is not trivial, hence infinite.
Thus, in all cases 1 and 2 are infinite additive subgroups of R:
As is injective, we observe that 1 \ 2 D ker D f0g: Hence, maps 2 injectively to
H \ .T f1g/: Since 2 is infinite, it follows that H \ .T f1g/ must be infinite. By Lemma
7.3 it follows that H \ .T f1g/ is dense in T f1g:
Likewise, H \ .f1g T/ is dense in f1g T: Let z D .s; t / 2 T: Then there exists a sequence
xn in H \ .T f1g/ with limit .s; 1/: Similarly, there exists a sequence yn in H \ .f1g T /
converging to .1; t /: It follows that xn yn is a sequence in H with limit z: Hence, H is dense.
We finally come to our example.
Example 7.5 We consider the group G D R2 =Z2 : The canonical projection W R2 ! R2 =Z2 is
a homomorphism of Lie groups. We recall from Example 4.17 that is a local diffeomorphism.
Accordingly, we use its tangent map to identify R2 D T0 R2 with g: Let X 2 R2 I then the
associated one-parameter subgroup D X in G is given by
.t / D tX C Z2 ; .t 2 R/:
From Lemma 7.2 it follows that the image H of X is a Lie subgroup of G: If X D 0; then X is
constant, and its image is the trivial group. We now assume that X 0: If X1 ; X2 have a rational
ratio, and X1 0; then X2 D pX1 =q; with p; q 2 Z; q > 0: Hence qX1 1 X 2 Z2 ; and it follows
that X is not injective. In the proof of Lemma 7.2 we saw that H is a compact submanifold of
G; diffeomorphic to the circle. A similar assertion holds in case X1 =X2 2 Q:
If X1 ; X2 have an irrational ratio, then tX Z2 for all t 2 R; so that X is injective. From
Corollary 7.4 it follows that H is dense in G in this case.
Lemma 7.6 Let ' W H ! G be an injective homomorphism of Lie groups. Then ' is immersive
everywhere. In particular, the tangent map ' D Te ' W h ! g is injective.
Proof: We will first establish the last assertion. There exists an open neighborhood of 0 in h
such that expH maps diffeomorphically onto an open neighborhood of e in H: The following
diagram commutes:
'
H ! G
expH " " expG
'
h ! g
Since expH is injective on ; it follows that ' expH is injective on I hence so is expG ' :
It follows that ' is injective on : Hence ker.' / \ D f0g: But ker.' / is a linear subspace
of hI it must be trivial, since its intersection with an open neighborhood of 0 is a point.
29
We have shown that ' is immersive at e: We may complete the proof by homogeneity. Let
h 2 H be arbitrary. Then l'.h/ ' lh 1 D ': Hence, by taking tangent maps at h it follows that
Th ' is injective.
In the following we assume that H is a Lie subgroup of G: The inclusion map is denoted by
W H ! G: As usual we denote the Lie algebras of these Lie groups by h and g; respectively.
The following result is an immediate consequence of the above lemma.
We recall that is a homomorphism of Lie algebras. Thus, via the embedding the Lie algebra
h may be identified with a Lie subalgebra of g; i.e., a linear subspace that is closed under the Lie
bracket. We will make this identification from now on. Note that after this identification the map
of the above diagram becomes the inclusion map.
h D fX 2 g j 8t 2 R W expG .tX/ 2 H g:
Proof: We denote the set on the right-hand side of the above equation by V:
Let X 2 h: Then expG .tX/ D .expH tX/ by commutativity of the above diagram with
' D : Hence, expG .tX/ 2 .H / D H for all t 2 R: This shows that h V:
To prove the converse inclusion, let X 2 g; and assume that X h: We consider the map
' W R h ! G defined by
'.t; Y / D exp.tX/ exp.Y /:
The tangent map of ' at .0; 0/ is the linear map T.0;0/ ' W R h ! g given by
Since X h; its kernel is trivial. By the immersion theorem there exists a constant > 0 and
an open neighbourhood of 0 in h; such that ' maps ; injectively into G: Shrinking
if necessary, we may in addition assume that expH maps diffeomorphically onto an open
neighborhood U of e in H:
The map m W .x; y/ 7! x 1 y; H H ! H is continuous, and maps .e; e/ to e: Since U
is an open neighborhood of e in H; there exists an open neighborhood U0 of e in H such that
m.U0 U0 / U; or, written differently,
U0 1 U0 U:
Since H is a union of countably many compact sets, there exists a countable collection fhj j j 2
Ng H such that the open sets hj U0 cover H: For every j 2 N we define
Tj D ft 2 R j exp tX 2 hj U0 g:
Let now j 2 N be fixed for the moment, and assume that s; t 2 Tj ; js t j < : Then it
follows from the definition of Tj that exp.t s/X D exp. sX/ exp.tX/ 2 U0 1 U0 U:
30
Hence exp.t s/X D exp Y for a unique Y 2 ; and we see that '.t s; 0/ D '.0; Y /: By
injectivity of ' on ; it follows that Y D 0 and s D t: From the above we conclude
that different elements s; t 2 Tj satify js t j : Hence Tj is countable.
The union of countably many countable sets is countable. Hence the union of the sets Tj is
properly contained in R and we see that there exists a t 2 R such that t Tj for all j 2 N: This
implies that exp tX [j 2N hj U0 D H: Hence X V: Thus we see that g n h g n V and it
follows that V h:
Example 7.8 Let V be a finite dimensional linear space (with k D R or C). In Example
2.14 we saw that SL.V / is a submanifold of GL.V /; hence a Lie subgroup. The Lie algebra of
GL.V / is equal to gl.V / D End.V /; equipped with the commutator brackets. We recall from
Example 2.14 that det W GL.V / ! k is a submersion at I: Hence the tangent space sl.V / of
SL.V / D det 1 .1/ at I is equal to ker.TI det/ D ker tr: We conclude that the Lie algebra of
SL.V / is given by
sl.V / D fX 2 End.V / j trX D 0gI (8)
in particular, it is a subalgebra of gl.V /: The validity of (8) may also be derived by using the
methods of this section, as follows.
If X 2 sl.V /; then by Lemma 7.7, exp.tX/ 2 SL.V / for all t 2 R; hence
d tX d
trX D det.e / D 1 D 0:
dt t D0 dt t D0
It follows that sl.V / is contained in the set on the right-hand side of (8).
For the converse inclusion, let X 2 End.V /; and assume that trX D 0: Then for every t 2 R
we have dete tX D e tr.tX/ D 1; hence exp tX D e tX 2 SL.V /: Using Lemma 7.7 we conclude
that X 2 sl.V /:
Example 7.9 We consider the subgroup O.n/ of GL.n; R/ consisting of real n n matrices x
with x t x D I: Being a closed subgroup, O.n/ is a Lie subgroup. We claim that its Lie algebra is
given by
o.n/ D fX 2 M.n; R/ j X t D Xg; (9)
the space of anti-symmetric n n matrices. Indeed, let X 2 o.n/: Then by Lemma 7.7, exp sX 2
O.n/; for all s 2 R: Hence,
t
I D .e sX /t e sX D e sX e sX :
Differentiating with respect to s at s D 0 we obtain X t C X D 0; hence X belongs to the set on
the right-ghand side of (9).
For the converse inclusion, assume that X 2 M.n; R/ and X t D X: Then, for every s 2 R;
t
.e sX /t e sX D e sX e sX D e sX sX
e D I:
31
SL.n; R/: We conclude that O.n/e SO.n/ O.n/: Since SO.n/ is connected, see exercises,
it follows that
O.n/e D SO.n/:
The determinant det W O.n/ ! R has image f 1; 1g and kernel SO.n/; hence induces a group
isomorphism O.n/=O.n/e ' f 1; 1g: It follows that O.n/ consists of two connected compo-
nents, O.n/e and xO.n/e ; where x is any orthogonal matrix with determinant 1: Of course,
one may take x to be the diagonal matrix with 1 in the bottom diagonal entry, and 1 in the
remaining diagonal entries, i.e., x is the reflection in the hyperplane xn D 0:
Lemma 7.10 Let G be a Lie group and H G a subgroup. Then H allows at most one
structure of Lie subgroup.
Theorem 7.11 Let G be a Lie group with Lie algebra g: If h g is a Lie subalgebra, then
the subgroup hexp hi generated by exp h has a unique structure of Lie subgroup. Moreover, the
map h 7! hexp hi is a bijection from the collection of Lie subalgebras of g onto the collection of
connected Lie subgroups of G:
Remark 7.12 In the literature, the group hexp hi is usually called the analytic subgroup of G
with Lie algebra h:
for every m 2 M: If is any such neighborhood, then also Tm M D Te .rm /h for all m 2 M:
In the literature one usually proves this result by using the Frobenius integrability theorem
for subbundles of the tangent bundle. We will first recall this proof, and then give an independent
proof based on a calculation of the derivative of the exponential map.
32
Proof: We consider the subbundle S of T G given by Sx D Te .lx /: Then for all X 2 h the
left invariant vector field vX is a section of S: We note that vX ; vY D vX;Y (see one of the
exercises). Hence for all X; Y 2 h the Lie bracket of vX and vY defines a section of S as well.
Let now X1 ; : : : ; Xk and put vj D vXj :
P P
Let now ; be any pair of smooth sections of SI then D kiD1 i vi and D kjD1 j vj
for uniquely defined smooth functions i and j on G: Since
X
; D i j vi ; vj C i vi .j /vj C j vj . i /vi ;
i;j
it follows that ; is a section of S: By the Frobenius integrability theorem it follows that the
bundle S is integrable. In particular, there exists a k-dimensional submanifold N of G containing
e; such that Tx N D Sx for all x 2 N:
For X 2 h; the vector field vX is everywhere tangent to N; hence restricts to a smooth vector
N
field vX on N: By smooth parameter dependence of this vector field on X there exists an open
neighborhood U of 0 in h and a positive constant > 0 such that for every X 2 U the integral
N
curve
X of vX with initial point e is defined on I WD ; : This integral curve is also an
integral curve for vX ; hence equals X W t 7! exp tX on I : It follows that exp U N: We
may now select an open neighborhood of 0 in g such that exp is a diffeomorphism from
onto an open subset of G and such that \ h is contained in U: Then M WD exp. \ h/ is
a k-dimensional hence open submanifold of N: It follows that Tx M D Tx N D Te .lx /h for all
x 2 M:
For the last assertion, we note that rm D rm lm1 lm so that rm D lm Cm 1 and Te .rm / D
Te .lm /Ad.m 1 /; and it suffices to show that Ad.m 1 / leaves h invariant. Write m D exp X with
X 2 \ h: Then
Ad.m 1 / D Ad.exp. X// D e adX
and the result follows, since ad.X/ leaves the closed subspace h invariant.
We shall now give a different proof of Lemma 8.1. The following result plays a crucial role.
33
From this it follows by interchanging partial derivatives, that
@ @ @
F .sX; sY / D exp. sX/ exp.sX C t sY /:
@s @t @s
t D0
Now,
@ @
exp. sX/ exp.sX C t sY / D exp. . s C /X/ exp..s C /.X C t Y //
@s @ D0
@
D exp. sX/ exp. X/ exp..X C t Y // exp.sX C t sY /
@ D0
@
D Te .lexp. sX/ rexp.sXCst Y / / exp. X/ exp.X C t Y /
@ D0
D Te .lexp. sX/ rexp.sXCst Y / /.t Y /;
Remark 8.3 The integral in the above expression may be expressed as a power series as follows.
Let V be a finite dimensional linear space, and A 2 End.V /: Then using the power series
expansion for e sA ; we obtain
Z 1 X1 Z 1
sA 1 n n
e ds D A s ds
0 nD0 0 n
1
X 1
D An :
nD0
.n C 1/
34
For obvious reasons, the sum of the latter series is also denoted by .e A I /=A:
Alternative proof of Lemma 8.1: Let be an open neighborhood of 0 in g such that exp j is
a diffeomorphism onto an open subset of G: Then M WD exp.h \ / is a smooth submanifold
of G of dimension dimM D dimh: For (10), we put m D exp X; with X 2 h \ : Since h is a
subalgebra, .e adX I /= adX leaves h invariant. Hence
I e adX
Tm M D TX .exp/h D Te .lm / h Te .lm /h:
adX
Equality follows for dimensional reasons. The identity with Te rm is proved in a similar manner.
We shall now proceed with the proof of Theorem 7.11, starting with the the result of Lemma
8.1, with and M as given there.
Lemma 8.4 Let C be a compact subset of M: Then there exists an open neighborhood U of 0
in g such that m exp.h \ U / is open in M for all m 2 C: In particular, C exp.h \ U / is an open
neighborhood of C in M:
Corollary 8.5 Let M G be as in Lemma 8.1. Then for every x1 ; x2 2 G; the intersection
x1 M \ x2 M is open in both x1 M and x2 M:
35
Proof of Theorem 7.11. Let H be the group generated by exp h: We will first equip H with the
structure of a manifold.
We fix and M as in Lemma 8.1. Replacing by a smaller neighborhood if necessary, we
may assume that exp j is a diffeomorphism of onto an open subset of G: Then exp restricts
to a diffeomorphism of 0 WD \ h onto the submanifold M of G: Accordingly, its inverse
W M ! 0 is a diffeomorphism of manifolds as well.
Since M H; it follows that H is covered by the submanifolds hM of G; where h 2 H:
We equip H with the finest topology that makes the inclusions hM ,! H continuous. Then
by definition a subset U H is open if and only if U \ hM is open in hM for every h 2 H:
We note that by Corollary 8.5, each set hM , for h 2 H; is open in H: For each open subset
O G and each h 2 H; the set O \ hM is open in hM I hence O \ H is open in H: It follows
that the inclusion map H ,! G is continuous. Since G is Hausdorff, it now follows that H;
with the defined topology, is Hausdorff. For each h 2 H; the map h D lh 1 W hM ! 0
is a diffeomorphism. This automatically implies that the transition maps are smooth. Hence
fh j h 2 H g is an atlas.
Fix a compact neighborhood C0 of 0 in \ h: Then C D exp C0 is a compact neighborhood
of e in M: It follows that C is compact in H: Since h is the union of the sets nC0 for n 2 N; it
follows that exp h is the union of the sets fc n j c 2 C g; for n 2 N: One now readily sees that
H is the union of the sets C n ; for n 2 N: Each of the sets C n is compact, being the image of
the compact Cartesian product C C (n factors) under the continuous multiplication map.
Hence the manifold H is a countable union of compact subsets, which in turn implies that its
topology has a countable basis.
We will finish the proof by showing that H with the manifold structure just defined is a Lie
group. If h 2 H then the map lh W H ! H is a diffeomorphism by definition of the atlas. We
will first show that right multiplication rh W H ! H is a diffeomorphism as well.
If X 2 h then the linear endomorphism Ad.exp X/ W g ! g equals e adX hence leaves h
invariant. Since H is generated by elements of the form exp X with X 2 h; it follows that
for every h 2 H the linear endomorphism Ad.h/ of g leaves h invariant. Fix h 2 H: Then
there exists an open neighborhood O of 0 in g such that Ad.h 1 /.O/ hence also
Ad.h 1 /.h \ O/ h \ : From exp Xh D h exp Ad.h 1 /X we now see that
1
h rh D Ad.h/ e on exp.h \ O/:
36
Replacing U by its intersection with ; we see that N0 D exp.h \ U / is an open neighborhood
of e in M and that Ne N0 M: It follows that the smooth map G maps Ne N0 into M; hence
its restriction G jNe N0 ; which equals H jNe N0 ; maps Ne N0 smoothly into the smooth
submanifold M of G: This implies that H is smooth in an open neighborhood of .e; e/ in
H H:
For the inversion, we note that 1 WD \ . / is an open neighborhood of 0 in g that
is stable under reflection in the origin. It follows that G maps the open neighborhood N1 WD
exp.1 \ h/ of e in M into itself. Hence its restriction to N1 ; which equals H jN1 ; maps N1
smoothly into the smooth manifold M: It follows that H is smooth in an open neighborhood of
e:
9 Closed subgroups
Theorem 9.1 Let H be a subgroup of a Lie group G: Then the following assertions are equiv-
alent:
(a) H is a C 1 -submanifold of G at the point eI
(b) H is a C 1 -submanifold of GI
(c) H is a closed subset of G:
Note that condition (b) implies that H is a Lie subgroup of G: Indeed, the map H W H !
H; h 7! h 1 is the restriction of the smooth map G to the smooth manifold H; hence smooth.
Similarly H is the restriction of G to the smooth submanifold H H of G G; hence smooth.
In the proof of the theorem we will need the following result. If G is a Lie group we shall
use the notation log for the map G ! g; defined on a sufficiently small neighborhood U of e;
that inverts the exponential map, i.e., exp log D I on U:
Proof: Being the local inverse to exp; the map log is a local diffeomorphism at e: Its tangent
map at e is given by Te log D .T0 exp/ 1 D Ig :
The map W g g ! G; .X; Y / 7! exp X exp Y has tangent map at .0; 0/ given by
T.0;0/ W .X; Y / 7! X C Y: The composition log is well defined on a sufficiently small
neighborhood of .0; 0/ in g g: Moreover, by the chain rule its derivative at .0; 0/ is given by
.X; Y / 7! X C Y:
It follows that, for .X; Y / 2 g g sufficiently close to .0; 0/;
37
where .X; Y / D o.kXk C kY k/ as .X; Y / ! .0; 0/ (here k k is any choice of norm on g).
Hence
Proof of Theorem 9.1. Let n D dimG: We first show that .a/ ) .b/: Let k be the dimension of
H at e: By the assumption there exists an open neighborhood U of e in G and a diffeomorphism
onto an open subset of Rn ; such that .U \ H / D .U / \ Rk f0g: Let now h 2 H: Then
Uh WD hU D lh .U / is an open neighborhood of h in G; and h WD lh 1 is a diffeomorphism
from Uh onto .U /: Moreover, since hU \ H D h.U \ H /; it follows that h .Uh \ H / D
.U \ H / D .U / \ Rk f0g: It follows that H is a C 1 -submanifold at any of its points.
Next we show that .b/ ) .c/: Assume (b). Then there exists an open neighborhood U of e
in G such that U \ HN D U \ H: Let y 2 HN : Then yU is an open neighborhood of y 2 HN in G;
hence there exists a h 2 yU \ H: Hence y 1 h 2 U: On the other hand, from y 2 HN ; h 2 H it
follows that y 1 h 2 HN : Hence y 1 h 2 U \ HN D U \ H; and we see that y 2 H: We conclude
that HN H; hence H is closed.
Finally, we show that .c/ ) .a/: We call an element X 2 g tangential to H if there exist
sequences Xn 2 g; n 2 R such that limn!1 Xn D 0; exp Xn 2 H; and limn!1 n Xn D X:
Let T be the set of X 2 g that are tangential to H: From the definition it is obvious that for all
X 2 T we have RX T:
We claim that for every X 2 T we have exp X 2 H: Indeed, let X 2 T; and let Xn ; n be
sequences as above. If X D 0 then obviously exp X 2 H: If X 0; then jn j ! 1: Choose
mn 2 Z such that n 2 mn ; mn C 1: Then mn ! 1 hence
38
origins in S and T respectively such that ' is a diffeomorphism from S T onto an open
neighborhood U of e in G:
We will finish the proof by establishing the claim that for S and T sufficiently small, we
have
'.f0g T / D U \ H:
Assume the latter claim to be false. Fix decreasing sequences of neighborhoods kS and kT of
the origins in S and T ; respectively, with kS kT ! f0g: By the latter assertion we mean
that for every open neighborhood O of .0; 0/ in S T; there exists a k such that kS kT S:
By the assumed falseness of the claim, we may select hk 2 '.kS kT / \ H such that hk
'.f0g kT /; for all k:
There exist unique Xk 2 kS and Yk 2 kT such that hk D '.Xk ; Yk / D exp Xk exp Yk :
From the above it follows that Xk is a sequence in S n f0g converging to 0: Moreover, from
exp Xk D hk exp. Yk / we see that exp Xk 2 H for all k: Fix a norm k k on S: Then the
sequence Xk =kXk k is contained in the closed unit ball in S; which is compact. Passing to a
suitable subsequence we may arrange that kXk k 1 Xk converges to an element X 2 S of norm
1: Applying the definition of T with n D kXk k 1 ; we see that also X 2 T: This contradicts the
assumption that S \ T D f0g:
Corollary 9.3 Let G; H be Lie groups, and let ' W G ! H be a continuous homomorphism of
groups. Then ' is a C 1 -map (hence a homomorphism of Lie groups).
Proof: Let D f.x; '.x// j x 2 Gg be the graph of ': Then obviously is a subgroup of the
Lie group G H: From the continuity of ' it follows that is closed. Indeed, let .g; h/ belong
to the closure of in G H: Let
n D .gn ; hn / be a sequence in converging to .g; h/: Then
gn ! g and hn ! h as n ! 1: Note that hn D '.gn /: By the continuity of ' it follows that
hn D '.gn / ! '.g/: Hence '.g/ D h and we see that .g; h/ 2 : Hence is closed.
It follows that is a C 1 -submanifold of G H: Let p1 W G H ! G and p2 W G H ! H
the natural projection maps. Then p D p1 j is a smooth map from the Lie group onto G: Note
that p is a bijective Lie group homomorphism with inverse p 1 W g 7! .g; '.g//: Thus p 1 is
continuous. By the lemma below p is a diffeomorphism, hence p 1 W G ! is C 1 : It follows
that ' D p2 p 1 is a C 1 -map.
39
Replacing G by a smaller neigborhood if necessary we may assume that p.UG / UH (use
continuity of p). Since p is a homeomorphism, p.UG / is an open subset of UH ; containing e:
Thus 0H WD .expH / 1 .p.UG //\H is an open neighborhood of 0 in h; contained in H : Note
that expH is a diffeomorphism from 0H onto UH0 D p.UG / UH : From the commutativity of
the diagram and the bijectivity of expG W G ! UG ; expH W 0H ! UH0 and p W UG ! UH0
it follows that p is a bijection of G onto 0H : It follows from this that p is a bijective linear
map. Its inverse p 1 is linear, hence C 1 : Lifting via the exponential maps we see that p 1 maps
UH0 smoothly onto UG I it follows that p 1 is C 1 at e: By homogeneity it follows that p 1 is C 1
everywhere. Indeed, let h 2 H: Then p 1 D lp.h/ p 1 lh 1 ; since p 1 is a homomorphism.
But lh 1 maps hUH0 smoothly onto UH0 I hence p 1 is C 1 on hUH0 :
X D X; trX D 0:
From this one sees that as a real linear space su.2/ is generated by the elements
i 0 0 1 0 i
r1 D ; r2 D ; r3 D :
0 i 1 0 i 0
Note that rj D i j ; where 1 ; 2 ; 3 are the famous Pauli spin matrices. One readily verifies
that r21 D r22 D r23 D I and r1 r2 D r2 r1 D r3 ; and r2 r3 D r3 r2 D r1 :
Remark 10.1 One often sees the notation i D r1 ; j D r2 ; k D r3 : Indeed, the real linear span
H D RI Ri Rj Rk is a realization of the quaternion algebra. The latter is the unique (up to
isomorphism) associative R algebra with unit, on the generators i; j; k; subject to the above well
known quaternionic relations.
40
It follows from the above product rules that the commutator brackets are given by
From this it follows that the endomorphisms adrj 2 End.su.2// have the following matrices
with respect to the basis r1 ; r2 ; r3 W
0 1 0 1 0 1
0 0 0 0 0 2 0 2 0
@
mat adr1 D 0 0 A
2 ; mat adr2 D @ 0 A @
0 0 ; mat adr3 D 2 0 0 A:
0 2 0 2 0 0 0 0 0
Lemma 10.2 Let t 2 R: Then exp tRa is the rotation with axis a and angle t jaj:
1
Proof: Let r 2 SO.3/: Then one readily verifies that Ra D r Rr 1a r ; and hence
1
exp tRa D r exptRr 1a r :
Selecting r such that r 1 a D jaje1 ; we see that we may reduce to the case that a D jaje1 : In that
case one readily computes that:
0 1
1 0 0
exp tRa D @ 0 cos t jaj sin t jaj A :
0 sin t jaj cos t jaj
Write Rj D Rej ; for j D 1; 2; 3: Then by the above formulas for mat ad.rj / we have
We now define the map ' W SU.2/ ! GL.3; R/ by '.x/ D matAd.x/; the matrix being taken
with respect to the basis r1 ; r2 ; r3 : Then ' is a homomorphism of Lie groups.
41
Proposition 10.3 The map ' W SU.2/ ! GL.3; R/; x 7! matAd.x/ is a surjective group
homomorphism onto SO.3/; and induces an isomorphism:
SU.2/=fI g ' SO.3/:
Proof: From
'.exp X/ D mate adX D e mat adX
we see that ' maps SU.2/e into SO.3/: Since SU.2/ is obviously connected, we have SU.2/ D
SU.2/e ; so that ' is a Lie group homomorphism from SU.2/ to SO.3/: The tangent map of
' is given by ' W X 7! mat adX: It maps the basis frj g of su.2/ onto the basis f2Rj g of
so.3/; hence is a linear isomorphism. It follows that ' is a local diffeomorphism at I; hence
its image im' contains an open neighborhood of I in SO.3/: By homogeneity, im' is an open
connected subgroup of SO.3/; and we see that im' D SO.3/e : As SO.3/ is connected, it follows
that im' D SO.3/: From this we conclude that ' W SU.2/ ! SO.3/ is a surjective group
homomorphism. Hence SO.3/ ' SU.2/= ker ': The kernel of ' may be computed as follows.
If x 2 ker '; then Ad.x/ D I: Hence xrj D rj x for j D 1; 2; 3: From this one sees that
x 2 f I; I g: Hence ker ' D f I; I g:
It is of particular interest to understand the restriction of ' to one-parameter subgroups of
SU.2/: We first consider the one-parameter group W t 7! exp.t r1 /: Its image T in SU.2/
consists of the matrices it
e 0
ut D ; .t 2 R/:
0 e it
Obviously, T is the circle group. The image of ut under ' is given by
'.ut / D '.e t r1 / D e ' .t r1 / D e 2tR1 :
By a simple calculation, we deduce that, for 2 R;
0 1
1 0 0
R WD e R1 D @ 0 cos sin A ;
0 sin cos
the rotation with angle around the x1 -axis. Let D be the group consisting of these rotations.
Then ' maps T onto D: Moreover, from '.ut / D R2t we see that ' restricts to a double covering
from T onto D:
More generally, if X is any element of su.2/; different from 0; there exists a x 2 SU.2/ such
that Ad.x 1 /X D x 1 Xx D r1 ; for some > 0: It follows that the one-parameter subgroup
X has image exp RX D expAd.x/Rr1 D xT x 1 in SU.2/: The image of xT x 1 under '
equals rDr 1 ; with r D '.x/: Moreover, the following diagram commutes:
Cx
T ! xT x 1
# ' # '
Cr 1
D ! rDr :
The horizontal arrows being diffeomorphisms, it follows that 'jxT x 1 is a double covering from
xT x 1 onto rDr 1 :
42
11 Group actions and orbit spaces
Definition 11.1 Let M be a set and G a group. A (left) action of G on M is a map W G M !
M such that
(a) .g1 ; .g2 ; m// D .g1 g2 ; m/ .m 2 M; g1 ; g2 2 G/I
(b) .e; m/ D m .m 2 M /:
Instead of the cumbersome notation we usually exploit the notation g m or gm for .g; m/:
Then the above rules (a) and (b) become: g1 .g2 m/ D .g1 g2 / m; and e m D m:
If g 2 G; then we sometimes use the notation g for the map m 7! .g; m/ D gm; M !
M: From (a) and (b) we see that g is a bijection with inverse map equal to g 1 : Let Sym.M /
denote the set of bijections from M onto itself. Then Sym.M /; equipped with the composition
of maps, is a group. According to (a) and (b) the map W g 7! g is a group homomorphism of
G into Sym.M /: Conversely, any group homomorphism G ! Sym.M / comes from a unique
left action of G on M in the above fashion.
Let M1 ; M2 be two sets equipped with (left) G-actions. A map ' W M1 ! M2 is said to
intertwine the G-actions, or to be equivariant, if '.gm/ D g'.m/ for all m 2 M1 and g 2 G:
Our goal is to study smooth actions of a Lie group on a manifold. As a first step we concen-
trate on continuous actions. This is most naturally done for topological groups.
Definition 11.3 A topological group is a group G equipped with a topology such that the mul-
tiplication map W G G ! G; .x; y/ 7! xy and the inversion map W G ! G; x 7! x 1 are
continuous.
We assume that H is a topological group and that M is a topological space equipped with
a continuous right action of H: Given h 2 H we denote by h the map M ! M given by
m 7! mh: Then h is continuous and so is its inverse h 1 : Therefore, h is a homeomorphism
of M onto itself.
43
Sets of the form mH .m 2 M / are called orbits for the action : Note that for two orbits
m1 H; m2 H either m1 H D m2 H or m1 H \ m2 H D ;: Thus, the orbits constitue a partition
of M: The set of all orbits, called the orbit space, is denoted by M=H: The canonical projection
M ! M=H; m 7! mH is denoted by :
The orbit space X D M=H is equipped with the quotient topology. This is the finest topology
for which the map W M ! M=H is continuous. Thus, a subset O of X is open if and only if
its preimage 1 .O/ is open in M:
In general this topology need not be Hausdorff even if M is Hausdorff. We will return to this
issue later.
The following result is useful, but particular for group actions. It is not true for quotient
topologies in general.
Lemma 11.5 The natural map W M ! M=H is open.
Proof: Let U M be open and put O D .U /: Then the preimage 1 .O/ equals the union
of the sets U h D h .U /; which are open in M: It follows that 1 .O/ is open, hence O is open
by definition of the quotient topology.
We denote by F .M / the complex linear space of functions M ! C: Let F .M /H denote the
subspace of F .M / consisting of functions g W M ! C that are H -invariant, i.e., g.mh/ D g.m/
for all m 2 M; h 2 H:
If f W M=H ! C is a function, then the pull-back of f by ; defined by .f / WD f ; is
a function on M that is H -invariant, i.e., it belongs to F .M /H : One readily verifies that is a
linear isomorphism from F .M=H / onto F .M /H :
Let C.M /H be the space C.M / \ F .M /H of continuous functions M ! C which are
H -invariant.
Lemma 11.6 The pull-back map W f 7! f maps C.M=H / bijectively onto C.M /H :
Proof: Obviously maps C.M=H / injectively into C.M /H : It remains to establish surjectiv-
ity. Let f 2 C.M /H : Then f D .g/ for a unique function g W M=H ! C: We must show
that g is continuous. Let be an open subset of C: Then U D f 1 ./ is open in M: From the
H -invariance of f it follows that U is right H -invariant. Hence U D 1 ..U // and it follows
that .U / is open in M=H: But .U / D g 1 ./: Thus, g is continuous.
Remark 11.7 With exactly the same proof it follows: if X is an arbitrary topological space, then
maps C.M=H; X/ (bijectively) onto C.M; X/H : In fact, the quotient topology on M=H is
uniquely characterized by this property for all X:
In what follows we shall mainly be interested in actions on locally compact Hausdorff spaces.
Recall that the topological space M is said to be Hausdorff if for each pair of distinct points
m1 ; m2 of M there exist open neighborhoods Uj of mj such that U1 \ U2 D ;: The space
M is said to be locally compact if each point in M has a compact neighborhood. Note that
in a Hausdorff space M each compact subset is closed. Moreover, if M is locally compact
Hausdorff, then for every point m 2 M and every open neighborhood U of m there exists a
compact neighborhood N of m contained in U:
44
Lemma 11.8 Let M be a locally compact Hausdorff space, equipped with a continuous right
action of a topological group H: Then the following assertions are equivalent.
(a) The orbit space M=H is Hausdorff.
(b) For each compact subset C M the set CH is closed.
Proof: Assume (a) and let C M be compact. Then .C / is compact. As M=H is Hausdorff,
it follows that .C / is closed. As CH D 1 ..C //; it follows that CH is closed.
Next, assume (b). From the fact that fmg is compact, for m 2 M; it follows that the orbit
mH is closed. Let x1 ; x2 2 X D M=H be distinct points. Select mj 2 1 .xj /: Then
xj D mj H; with m1 H \ m2 H D ;: The complement V of m2 H in M is open, right H -
invariant, and contains m1 H: Select an open neighborhood U1 of m1 in M such that UN 1 is
compact and contained in V: Then by (a), the set UN 1 H is closed and still contained in V: Its
complement V2 is open and containes m2 H: Hence, .V2 / is open in X and contains x2 :
On the other hand, V1 D U1 H is the union of the open sets U1 h; hence open in M: Moreover,
V1 contains m1 ; so that .V1 / is an open neighborhood of x1 in X: Clearly, the sets V1 and V2 are
right H -invariant and disjoint. It follows that the sets .V1 / and .V2 / are disjoint open subsets
of X containing the points x1 and x2 ; respectively. This establishes the Hausdorff property.
In the rest of this section we will always assume that M is a smooth manifold on which H
has a smooth right action. We will first study smooth actions for which the quotient M=H allows
a natural structure of smooth manifold.
If is a smooth manifold, then H has a right action on the manifold H; given by
.x; g/ h D .x; gh/: We will say that such an action is of trivial principal fiber bundle (or trivial
PFB) type.1
More generally, the right action of H on a manifold M is called of trivial PFB type if there
exist a smooth manifold and a diffeomorphism W M ! H that intertwines the H -
actions. Such a map is called a trivialization of the action. Note that dim D dimM dimH:
Definition 12.2 The right action of H on M is called of principal fiber bundle (PFB) type if the
following two conditions are fulfilled.
(a) Every point m of M possesses an open H -invariant neighborhood U such that the right
H -action on U is of trivial PFB type.
(b) If C is a compact subset of M; then CH is closed.
1
The terminology principal fiber bundle type is not standard, but used here for purposes of exposition.
45
In view of Lemma 11.8, the second condition is equivalent to the condition that the quotient
space M=H is Hausdorff.
We call the pair .U; / of condition (a) a local trivialization of the right H -space M at the
point m: Clearly, if the right H -space M is of PFB type, then there exists a collection f.U ; / j
2 Ag of local trivializations such that the open sets U cover M: Such a covering is called a
trivializing covering.
Remark 12.3 If H is a closed subgroup of a Lie group G; then the map .h; g/ 7! gh; H G !
G defines a smooth right action of H on G: At a later stage we will see that this action is of PFB-
type.
If the right H -action on M is of PFB type, then the quotient M=H admits a unique natural
structure of smooth manifolds. In order to understand the uniqueness, the following preliminary
result about submersions will prove to be very useful.
Proof: The map ; being a submersion, is open. In particular, .X/ is an open subset of Y: Let
y0 2 .X/: Fix x0 2 X such that .x0 / D y0 : Since is a submersion, there exists an open
neighborhood U of x0 and a diffeomorphism ' W U ! .U / F; with F a smooth manifold of
dimension dimX dimY; such that D pr2 ': Here pr2 denotes the projection .U / F ! F
onto the second component. Let b D pr2 '.x0 /: Then the smooth map W .U / ! X defined
by .y/ D ' 1 .y; b/ satisfies D I on .U / and .y0 / D x0 : In other words, admits a
smooth locally defined section with .y0 / D x0 : From this it follows that D D
' on .U /: Hence, is smooth on .U /:
Theorem 12.5 Let the right H action on M be of PFB type. Then M=H carries a unique
structure of C 1 -manifold (compatible with the topology) such that the canonical projection
W M ! M=H is a smooth submersion.
If m 2 M; then the tangent map Tm W Tm M ! T.m/ .M=H / has kernel Tm .mH /;
the tangent space of the orbit mH at m: Accordingly, it induces a linear isomorphism from
Tm M=Tm .mH / onto T.m/ .M=H /:
Finally, W f 7! f restricts to a bijective linear map from C 1 .M=H / onto C 1 .M /H :
Remark 12.6 It follows from the assertion on the tangent maps that the dimension of M=H
equals dimM dimH:
46
Proof: We will first show that the manifold structure, if it exists, is unique. Let Xj denote
M=H; equipped with a manifold structure labeled by j 2 f1; 2g and assume that the projection
map W M ! M=H is submersive for both manifold structures. We will show the identity map
I W X1 ! X2 is smooth for the given manifold structures. The following diagram commutes
M
. &
I
X1 ! X2
Since W M ! X1 is a submersion and W M ! X2 smooth, it follows by Lemma 12.4
that I W X1 ! X2 is smooth. By symmetry of the argument it follows that the inverse to I is
smooth as well. Hence, X1 and X2 are diffeomorphic manifolds. This establishes uniqueness of
the manifold structure.
We defer the treatment of existence until the end of the proof, and will first derive the other
assertions as consequences.
We first address the assertion about the tangent map of : Let m 2 M: Since is a submer-
sion, Tm W Tm M ! T.m/ .M=H / is a surjective linear map, with kernel equal to the tangent
space of the fiber 1 ..m//: This fiber equals mH: Hence ker Tm D Tm .mH /:
Finally, it is obvious that restricts to a linear injection from C 1 .M=H / into C 1 .M /H :
Let g 2 C 1 .M /H : Then g D f for a unique function f W M ! C: Since g is smooth
and a smooth submersion, it follows by application of Lemma 12.4 that f is smooth. This
establishes the surjectivity, and hence the bijectivity of :
We end the proof by establishing the existence of a manifold structure on X D M=H for
which becomes a smooth submersion. First of all, X is a topological space, which is Hausdorff
because of Lemma 11.8.
Let f.U ; /g2A be a trivializing covering of M as above. Thus, is a diffeomorphism of
U onto H which intertwines the right H -actions. Writing the manifolds as unions of
charts, we see that we may replace the trivializing covering by one for which each equals an
open subset of Rn : We write i for the injection x 7! .x; e/; ! H and p for the
projection H ! onto the first coordinate.
We will use the trivializing covering to define a smooth atlas of X: The map W U !
H is a diffeomorphism intertwining the H -actions, hence induces a homeomorphism W
.U / ! . H /=H: The projection map p induces a homeomorphism of the latter space
onto ; by which we shall identify. Put V D .U /: Then the following diagram commutes:
U ! H
# # p (14)
V !
The sets V ; for 2 A; constitute an open covering of X; and the maps W V ! are
homeomorphisms. We will show that the pairs .V ; /; for 2 A; constitute a smooth atlas.
Put D .V \ V /: Then the transition map
WD 1
47
is a homeomorphism from onto : We must show it is smooth.
The transition map D 1 is a diffeomorphism from H onto H: Moreover,
the diagram
H ! H
p # # p
!
commutes. As the vertical arrow represent smooth submersions, it follows by application of
Lemma 12.4 that is smooth.
Let X be equipped with the structure of C 1 -manifold determined by the atlas defined above.
The map maps U onto V : Moreover, from the commutativity of the diagram (14) we see that
jU corresponds via the horizontal diffeomorphisms and with the smooth projection p :
Hence is smooth and submersive on each U I it follows that is a smooth submersion.
The following terminology is standard in the literature, and explains the terminology PFB
type used so far. We assume that X is a smooth manifold.
Definition 12.7 A principal fiber bundle over X with structure group H is a pair .P; / consist-
ing of a smooth right H -manifold P and a smooth map W P ! X with the following property.
For every point x 2 X there exists an open neighborhood V of x in X and a diffeomorphism
W 1 .V / ! V H such that
1
(a) D prV on .V /; where prV denotes the projection V H ! V I
(b) intertwines the right H -actions.
The manifold P is called the total space, X is called the base space of the bundle. A map as
above is called a local trivialization of the bundle.
The terminology action of PFB type is finally justified by the following result.
Proof: Assertion (a) is a straightforward consequence of Definition 12.7. Assertion (b) is easily
seen from the proof of Theorem 12.5.
Example 12.9 (Frame bundle of a vector bundle) Let V be a finite dimensional real vector space
of dimension k: Let Hom.Rk ; V / denote the linear space of linear maps Rk ! V: A frame in
V is defined to be an injective linear map f W Rk ! V: The set of frames, denoted F .V /, is
a dense open subset of Hom.Rk ; V /: Let e1 ; : : : ; ek be the standard basis of Rk : Then the map
48
f 7! .f .e1 /; : : : ; f .ek // is a bijection from F .V / onto the set of ordered bases of V: Thus, a
frame may be specified by giving an ordered basis of V:
The group H WD GL.k; R/ ' GL.Rk / acts on F .V / from the right; indeed the action is
given by .f; a/ 7! f a: This action is free and transitive; see the text preceding Theorem 13.5
and Proposition 15.5 for the definitions of these notions. Thus, for each f 2 F .V / the map
a 7! f a is a diffeomorphism from H onto F .V /:
Let now p W E ! M be a vector bundle of rank k over a smooth manifold M: For an open
subset U M we write EU WD p 1 .U /: Then p W EU ! U is a vector bundle over U; called
the restriction of E to U: A trivialization of E over an open subset U M is defined to be an
isomorphism W EU ! U Rk of vector bundles. For x 2 U we define the linear isomorphism
x W Ex ! Rk by D .x; x / on Ex : Let RkM denote the trivial vector bundle M Rk over M:
Then the vector bundle Hom.RkM ; E/ has fiber Hom.Rk ; Ex / at the point x 2 M: A trivialization
.U; / of E induces a trivialization 0 of Hom.Rk ; E/ given by x0 .Tx / D x Tx for Tx 2
Hom.Rk ; Ex /:
We define F .E/ to be the subset [x2M F .Ex / of Hom.RkM ; E/: This subset is readily seen
to be open; the natural map F .E/ ! M mapping F .E/x to x defines a sub fiber bundle of
Hom.RkM ; E/: A trivialization of E over U induces a trivialization 00 of F .E/ over U given
by x00 .f / D x f for f 2 F .Ex /: The group H acts from the right on each fiber F .Ex /: By
looking at trivializations we see that these actions together constitute a smooth right action of H
on F .E/ which turns F .E/ into a principal fiber bundle with structure group H:
Definition 13.1 The action of H on M is called proper if .m; h/ 7! .m; mh/ is a proper map
M H ! M M:
Remark 13.2 Note that a continuous action of a compact (in particular of a finite) group is
always proper.
49
Proof: Let ' W M H ! M M; .m; h/ 7! .m; mh/: Assume (a) and let C1 ; C2 M be
compact sets. Then C1 C2 is compact, hence ' 1 .C1 C2 / is a compact subset of M H:
Now
' 1 .C1 C2 / D f.m; h/ j m 2 C1 ; mh 2 C2 g;
hence HC1 ;C2 D p2 .' 1 .C1 C2 //; with p2 denoting the projection M H ! H: It follows
that HC1 ;C2 is compact. Hence (b).
Now assume that (b) holds, and let C be a compact subset of M M: Then there exist compact
subsets C1 ; C2 M such that C C1 C2 : Now ' 1 .C / is a closed subset of ' 1 .C1 C2 /;
hence it suffices to show that the latter set is compact. The latter set is clearly closed; moreover,
it is contained in C1 HC1 ;C2 ; hence compact.
Remark 13.4 We leave it to the reader to verify that condition (b) is equivalent to the condition
that fh 2 H j C h \ C ;g be compact, for any compact set C M:
Theorem 13.5 Let M be a smooth manifold equipped with a smooth right H -action. Then the
following statements are equivalent.
(a) the action of H on M is proper and free;
(b) the action of H on M is of PFB type.
Lemma 13.7 (Slice Lemma). Let M be a smooth manifold equipped with a smooth right H
action which is proper and free. Then for each m 2 M there exists a smooth submanifold S of
M containing m such that the map .s; h/ 7! sh maps S H diffeomorphically onto an open
H -invariant neighborhood of m in M:
Remark 13.8 The manifold S is called a slice for the H -action at the point m:
50
Proof: Fix m 2 M and define the map m W H ! M by h 7! mh: By freeness of the action,
this map is injective. We claim that its tangent map at e is an injective linear map h ! Tm M:
Given X 2 h we define the smooth vector field vX on M by
d
vX .m/ D m exp tX:
dt t D0
vX .m/ D Te .m /.X/:
One readily sees that the integral curve of vX with initial point m is given by c W t 7! m exp tX:
From vX .m/ D 0 it follows that c is constant; by freeness of the action this implies that exp tX D
e for all t 2 R; hence X D 0: Thus vX .m/ D 0 ) X D 0 and it follows that the linear map
Te .m / has trivial kernel, hence is injective.
We now select a linear space s Tm M such that s Te .m /.h/ D Tm M: Moreover, we
select a submanifold S 0 of M of dimension dimM dimH which has tangent space at m equal
to s: Consider the map ' W S 0 H ! M; .s; h/ 7! sh: Then T.m;e/ ' W s h ! Tm M is
given by .X; Y / 7! X C Te .m /Y; hence bijective. Replacing S 0 by a neighborhood (in S 0 ) of
its point m we may as well assume that S 0 has compact closure and that there exists an open
neighborhood O of e in H such that ' maps S 0 O diffeomorphically onto an open subset of
M: In particular it follows that the tangent map T.s;e/ ' is injective for every s 2 S 0 : Using the
homogeneity h ' .I rh 1 / D ' for all h 2 H we see that ' has bijective tangent map at
every point of S 0 H:
Let C D HSN 0 ;SN 0 : Then C is a compact subset of H: Hence C0 D C n O is a compact subset
of H; not containing e: Note that m mC0 by freeness of the action. Hence there exists an open
subset S of S 0 containing m such that S \ Sh D ; for all h 2 C0 (use Lemma 13.6).
We claim that the map ' is injective on S H: Indeed, assume '.s1 ; h1 / D '.s2 ; h2 /;
for s1 ; s2 2 S; h1 ; h2 2 H: Then s2 D s1 .h1 h2 1 /; hence h1 h2 1 belongs to the compact set
C D HSN 0 ;SN 0 : From the definition of S it follows that h1 h2 1 2 C n C0 O: From the injectivity
of ' on S 0 O it now follows that s1 D s2 and h1 h2 1 D e: Hence ' is injective on S H:
Since we established already that ' has a bijective tangent map at every point of S H it now
follows that ' is a diffeomorphism from S H onto an open subset U of M: As '.m; e/ D m;
it follows that m 2 U: Moreover, ' intertwines the H -action on S H with the H -action on U:
Therefore, U is H -invariant.
Proof of Theorem 13.5. (a) ) (b): Assume (a). We shall first prove that the first condition of
Definition 12.2 holds. Let m 2 M and let S be a slice through m as in the above lemma. Then
the map ' W S H ! M given in the lemma is an H -equivariant diffeomorphism onto an H -
invariant open neighborhood U of m in M: It follows that the inverse map D ' 1 W U ! S H
is a trivialization of the H -action on U:
We now turn to the second condition of Definition 12.2. Let C M be compact and let x
be a point in the closure of CH: Fix a compact neighborhood C 0 of x in M: Then there exists a
sequence .xn /n1 in C 0 \ CH such that xn ! x as n ! 1: Write xn D cn hn ; with cn 2 C
51
and hn 2 H: Then hn is contained in HC;C 0 I the latter set is compact by condition (a). By
passing to subsequences if necessary, we arrive in the situation that the sequences .cn / and .hn /
are convergent, say with limits c 2 C and h 2 H; respectively. Now x D lim cn hn D ch 2 CH:
It follows CH contains its closure, hence is closed. This establishes the second condition of
Definition 12.2. Thus, (b) follows.
(b) ) (a): Assume (b) holds. To see that the action of H on M is free, let x 2 M; h 2 H
and assume that xh D x: There exists an H -invariant open neighborhood U of x on which the
H -action is of trivial PFB-type. Let W U ! H be a trivialization of the action. Then from
.xh/ D .x/ and .xh/ D .x/h it follows that .x/h D .x/: Hence h D e: This establishes
freeness of the action.
To see that the action of H on M is proper, let C; C 0 M be compact subsets. Then
it suffices to show that HC;C 0 D fh 2 H j C h \ C 0 ;g is compact. For every x 2 C
there exists an H -invariant open neighborhood Ux of x on which the action is of trivial type.
Moreover, there exists a compact neighborhood Cx of x contained in Ux : The interiors of the
sets Cx form an open cover of C; hence contain a finite subcover, parametrized by finitely many
elements x1 ; : : : ; xn 2 M: Put Ci D Cxi ; then C [niD1 Ci where Ci is contained in Uxi : One
easily verifies that H[i Ci ;C 0 D [i HCi ;C 0 : Therefore it suffices to prove that HC;C 0 is compact
under the assumption that C is contained in an H -invariant open set U on which the action is of
trivial type. Now CH is closed, hence C 00 D CH \C 0 is compact and contained in U: Moreover,
HC;C 0 D HC;C 00 : Thus, we may as well assume that C 0 U: Using a trivializing diffeomorphism
we see that we may as well assume that M is of the form H: Let D and D 0 be the projections
of C and C 0 onto H; respectively. Then D and D 0 are compact. Moreover, HC;C 0 is a closed
subset of fh 2 H j Dh \ D 0 ;g D D 1 D 0 : The latter set is the image of the compact set
D D 0 under the continuous map H H ! H; .h1 ; h2 / 7! h1 1 h2 ; hence compact. It follows
that HC;C 0 is compact as well. This establishes (a).
Example 13.9 We return to the setting of Example 12.9, with p W E ! M a rank k-vector
bundle. The frame bundle W F .E/ ! M is a principal fiber bundle with structure group H D
GL.k; R/: Thus, the action of H on F .E/ is proper and free, with quotient space F .E/=H ' M:
We observe that the bundle E can be retrieved from F .E/ as follows. The map ' W F .E/
Rk ! E defined by .f; v/ 7! f .v/ on F .E/x Rk ; for x 2 M; is a surjective smooth map.
Using trivializations of E one sees that ' is a submersion. Two elements .f1 ; v1 / and .f2 ; v2 /
have the same image if and only if they belong to the same fiber F .E/x Rk and there exists
a h 2 H such that .f2 ; v2 / D .f1 h; h 1 v1 /: Define the right action of H on F .E/ Rk by
.f; v/a D .f h; h 1 v/: Then it follows that the fibers of ' are precisely the orbits for the right
action of H on F .E/ Rk :
Via the projection q W F .E/ Rk ! F .E/ we view F .E/ Rk as a trivial vector bundle
over F .E/: The map q intertwines the given right actions of H: As the action of H on F .E/ is
proper and free, so is the action of H on F .E/ Rk (argument left to the reader). It follows
that the induced map qN W .F .E/ Rk /=H ! F .E/=H D M is smooth (show this). Using
trivializations of E, hence of F .E/; one readily checks that the projection qN defines a smooth
rank k vector bundle over M:
52
The map ' W F .E/Rk ! E defined above induces a smooth map 'N W .F .E/Rk /=H ! E
(give the argument). Again using trivializations of E one checks that 'N is an isomorphism
of vector bundles. Thus, the vector bundle qN W .F .E/ Rk /=GL.k; R/ ! M is naturally
isomorphic to E:
14 Coset spaces
We now consider a type of proper and free action that naturally occurs in many situations. Let
G be a Lie group and H a closed subgroup. The map .g; h/ 7! gh defines a smooth right action
of H on G: The associated orbit space is the coset space G=H; consisting of the right cosets
gH; g 2 G:
Lemma 14.1 Let H be a closed subgroup of the Lie group G: Then the right action of H on G
is proper and free.
Proof: It is clear that the action is free. To prove it is proper, let C1 ; C2 be compact subsets of
G: Then HC1 ;C2 D C1 1 C2 \ H: Now C1 1 C2 is the image of C1 C2 under the continuous map
.x; y/ 7! x 1 y; hence compact. Moreover, H is closed, hence HC1 ;C2 is compact.
Corollary 14.2 Let G be a Lie group and H a closed subgroup. Then the coset space G=H has
a unique structure of smooth manifold such that the canonical projection W G ! G=H is a
smooth submersion. Relative to this manifold structure, the following hold.
(a) The map W G ! G=H is a principal fiber bundle with structure group H:
(b) The left action of G on G=H given by .g; xH / 7! gxH is smooth.
Proof: From Lemma 14.1 and Theorem 13.5 it follows that the right action of H on G is of
PFB type. Hence, the first assertion is an immediate consequence of Theorem 12.5. Moreover,
assertion (a) follows from Lemma 12.8 (b). Finally, put X D G=H and let denote the action
map G X ! X: Then the following diagram commutes:
G G ! G
# I #
GX ! X:
Since the vertical map on the left side of the diagram is a submersion, whereas and are
smooth, it follows that is smooth (see Lemma 12.4).
Corollary 14.3 Let G be a Lie group and H a closed subgroup. The tangent map Te of
W G ! G=H is surjective and has kernel equal to h:
1
Proof: This is an immediate consequence of the fact that is a submersion with fiber .eH / D
H:
Remark 14.4 It follows from the above that the tangent map Te induces a linear isomorphism
from g=h onto TeH .G=H /I we agree to identify the two spaces via this isomorphism from now
on. With this identification, Te becomes identified with the canonical projection g ! g=h:
53
15 Orbits of smooth actions
In this section we assume that G is a Lie group and that M is a smooth manifold equipped with
a smooth left G-action :
Given X 2 g; we denote by X the smooth vector field on M defined by
d
X .m/ WD .exp tX/ m
dt t D0
We leave it to the reader to verify that for every m 2 M; the curve t 7! .exp tX/ m is the maximal
integral curve of X with initial point m:
Lemma 15.1 The map X 7! X is a Lie algebra anti-homomorphism from g into the Lie
algebra V.M / of smooth vector fields on M:
Proof: Fix m 2 M; and let m W G ! M; g 7! gm: Then X .m/ D Te .m /X: It follows that
X 7! X .m/ is a linear map g ! Tm M: This shows that X 7! X is a linear map g ! V.M /:
It remains to be shown that X ; Y D Y;X ; for all X; Y 2 g: Since .t; m/ 7! .exp tX/ m D
exp tX .m/ is the flow of X ; the Lie bracket of the vector fields X and Y is given by
d
X ; Y .m/ D Y .m/
dt t D0 exp tX
d
D T.exp tX/ m .exp tX /Y ..exp tX/ m/
dt t D0
d d
D .exp tX/.exp sY /.exp tX/ m
dt t D0 ds sD0
d d
D .exp se t adX Y /m
dt t D0 ds sD0
d
D e t adX Y .m/:
dt t D0
By linearity of Z 7! Z .m/ it follows from this that X ; Y .m/ D Z .m/; where Z D
.d=dt /e t adX Y jt D0 D X; Y :
Remark 15.2 Right multiplication x 7! rx defines a right action of G on itself. The associated
map g ! V.G/ is given by the map X 7! vX of Lemma 3.1 and defines a linear isomorphism of
g onto the space VL .G/ of left invariant vector fields on g: It follows from the above that VL .G/
is a Lie subalgebra of V.G/ and that X 7! vX is an isomorphism of Lie algebras from g onto
VL .G/:
If x 2 M; then the stabilizer Gx of x in G is defined by
Gx D fg 2 G j gx D xg:
Being the pre-image of x under the continuous map g 7! gx; the stabilizer is a closed subgroup
of G:
54
Lemma 15.3 Let x 2 M: The Lie algebra gx of Gx is given by
Proof: Let gx denote the Lie algebra of Gx : Then for all t 2 R we have exp tX 2 Gx ; hence
.exp tX/x D x: Differentiating this expression with respect to t at t D 0 we see that X .x/ D 0:
It follows that gx is contained in the set on the right-hand side of (15).
To establish the converse inclusion, assume that X .x/ D 0: Then c.t / D .exp tX/ x is
the maximal integral curve of the vector field X with initial point x: On the other hand, since
X .x/ D 0; the constant curve d.t / D x is also an integral curve. It follows that exp tX x D
c.t / D d.t / D x; hence exp tX 2 Gx for all t 2 R: In view of Lemma 7.7 it now follows that
X 2 gx :
As Gx is a closed subgroup of G; it follows from Corollary 14.2 that the coset space G=Gx
has the structure of a smooth manifold. Moreover, let W G ! G=Gx denote the canonical
projection. Then is a submersion, and the tangent space of G=Gx at eN WD .e/ is given by
TeN .G=Gx / ' g= ker Te D g=gx :
The map x W g 7! gx factors through a bijection N x of G=Gx onto the orbit Gx:
Proof: It follows from Corollary 14.2 that the natural projection W G ! G=Gx is a smooth
submersion. Since x D N x ; it follows by application of Lemma 12.4 that N x is smooth.
From x D N x it follows by taking tangent maps at e and application of the chain rule
that
Te x D TeN .N x / Te : (16)
Now Te is identified with the canonical projection g ! g=gx : Moreover, if X 2 g; then
Te .x /.X/ D d=dt x .exp tX/jt D0 D X .x/: Hence ker Te .x / D gx D ker Te : Combining
this with (16) we conclude that TeN N x is injective g=gx ! Tx M: Hence, N x is immersive at e:
N
We finish the proof by applying homogeneity. For g 2 G; let lg denote the left action of g on
G=Gx ; and let g denote the left action of g on M: Then the maps lg and g are diffeomorphisms
of G=Gx and M respectively, and
g N x lg 1 D N x :
By taking the tangent map of both sides at .g/ and applying the chain rule we may now con-
clude that N x is immersive at .g/:
The action of G on M is called transitive if it has only one orbit, namely the full manifold
M: In this case the G-manifold M is said to be a homogeneous space for G: The following result
asserts that all homogeneous spaces for G are of the form G=H with H a closed subgroup of G:
Proposition 15.5 Let the smooth action of G on M be transitive, and let x 2 M: Then the map
x W G ! M; g 7! gx induces a diffeomorphism G=Gx ' M:
55
Proof: The map N x W G=Gx ! M is a smooth immersion and a bijection. By Corollary 16.6
(see intermezzo on the Baire theorem) it must be a submersion at some point of G=Gx : By
homogeneity it must be a submersion everywhere. Hence N x is a local diffeomorphism. Since
N x is a bijection, we conclude that it is a diffeomorphism.
Example 15.6 Let n 0: The special orthogonal group SO.n C 1/ acts smoothly and naturally
on RnC1 : Let e1 be the first standard basis vector in RnC1 : Then the orbit SO.n C 1/e1 equals
the n-dimensional unit sphere S D S n in RnC1 : Since S is a smooth submanifold of RnC1 ; it
follows that the action of SO.n C 1/ on S is smooth and transitive. The stabilizer SO.n C 1/e1
equals the subgroup consisting of .n C 1/ .n C 1/ matrices of the form
1 0
; with B 2 SO.n/:
0 B
Example 15.7 Let n 0: We recall that n-dimensional real projective space P WD Pn .R/ is
defined to be the space of 1-dimensional linear subspaces of RnC1 : It has a structure of smooth
manifold, characterized by the requirement that the natural map W RnC1 n f0g ! P; v 7! Rv
is a smooth submersion.
We consider the natural smooth action of G WD GL.n C 1; R/ on RnC1 n f0g given by
.g; x/ 7! gx: Then G maps fibers of onto fibers, hence the given action induces an action
G P ! P: Since is a submersion, it follows by application of Lemma 12.4 that the action of
G on P is smooth. Let m 2 P be the line spanned by the first standard basis vector e1 of RnC1 :
Then Gm equals the group of invertible .n C 1/ .n C 1/ matrices with first column a multiple
of e1 : One readily sees that the action is transitive. Therefore, the induced map G=Gm ! P is a
diffeomorphism of manifolds.
We now consider the subgroup K D O.n C 1/ of G: One readily sees that K already acts
transitively on P: Hence the action induces a diffeomorphism from K=Km onto P: Here we note
that Km D K \ Gm consists of the matrices
a 0
;
0 B
with a D 1 and B 2 O.n/: Thus, Km ' O.1/ O.n/; and we see that
56
If O1 ; O2 are two open dense subsets of X; then O1 \O2 is still open dense. Indeed, if U X
is open non-empty, then U \ O1 is open non-empty by density of O1 : Hence, U \ O1 \ O2 is
open non-empty by density of O2 :
It follows that the intersection of finitely many open dense subsets of X is still open and
dense.
Definition 16.1 The topological space X is called a Baire space if every countable intersection
of open dense subsets is dense.
Theorem 16.3 (Baire category theorem) Let X be a Hausdorff topological space. Then X is a
Baire space as soon as one of the following two conditions if fulfilled.
(a) X is locally compact.
(b) There exists a complete metric on X that induces the topology of X:
57
Lemma 16.4 Let .X; d / be a complete metric space and let Ck be a decreasing sequence of
non-empty closed subsets of X whose diameters d.Ck / tend to zero. Then \k2N Ck consists of
precisely one point.
Proof: The condition about the diameter means that we may select a ball of radius rk containing
the set Ck ; for k 2 N; such that rk ! 0 as k ! 1: For each k we may select xk 2 Ck : Then
d.xm ; xn / < 2rk for all m; n k; hence .xn / is a Cauchy sequence. By completeness of the
metric, the sequence .xn / has a limit x:
Fix k 2 N: Let > 0; then there exists n k such that xn 2 B.xI /: Hence B.xI / \
Ck ;: It follows that x belongs to the closure of Ck ; hence to Ck ; for every k 2 N: Hence
x 2 \k2N Ck : If y is a second point in the intersection, then for every k; both x; y belong to Ck ;
hence d.x; y/ < 2rk : It follows that d.x; y/ D 0; hence x D y:
A useful application of the above is the following result.
Corollary 16.6 Let X and Y be smooth manifolds with dimX < dimY: Let ' W X ! Y be a
smooth immersion. Then '.X/ has empty interior.
Proof: Put d D dimX: For every x 2 X there exists an open neighborhood Ux of x in X such
that '.Ux / is a smooth submanifold of Y of dimension d: By the second countability assumption
there exists a countable covering of X by open subsets Uk of X such that '.Uk / is a smooth
submanifold of Y of dimension d: It follows that '.X/ D [k2N '.Uk / has empty interior.
Proposition 17.1 Let G be a Lie group and H a closed normal subgroup. Then G=H has a
unique structure of Lie group such that the canonical map W G ! G=H is a homomorphism
of Lie groups.
58
Proof: We equip G=H with the unique manifold structure for which is a submersion. Since
H is normal, G=H has a unique group structure such that is a group homomorphism. Let N
denote the multiplication map of the quotient group G=H: Then the following diagram commutes
GG ! G
# #
N
G=H G=H ! G=H
Since and are smooth, so is : Since the left vertical map is a submersion, it follows from
Lemma 12.4 that N is smooth. In a similar fashion it follows that the inversion map of G=H is
smooth. Hence G=H is a Lie group, and is a Lie group homomorphism.
Suppose that G=H is equipped with a second structure of Lie group such that W G ! G=H
is a Lie group homomorphism. We shall denote G=H; equipped with this structure of Lie group,
by .G=H /0 : The identity map I W G=H ! .G=H /0 clearly is an injective homomorphism of
groups. Since is a submersion, it follows by application of Lemma 12.4 that I is smooth,
hence a Lie group homomorphism. Since I is injective, it follows by Lemma 7.6 that I is
immersive everywhere. Hence, by Lemma 17.2 below we see that I is a submersion. Thus, I
is a bijective local diffeomorphism, hence a diffeomorphism. Therefore, I is an isomorphism of
Lie groups, establishing the uniqueness.
Lemma 17.2 Let ' W G ! G 0 be an immersive homomorphism of Lie groups. Then ' is a
submersion if and only if '.G/ is an open subgroup of G 0 :
Remark 17.3 In Proposition 17.7 we will see that the assumption that ' be immersive is super-
fluous.
Proof: If ' is a submersion, then '.G/ is open in G 0 : Conversely, assume that '.G/ is open in
G 0 : Then it follows by Corollary 16.6 that dimG D dimG 0 : Hence Te ' W Te G ! Te G 0 is an
injective linear map between spaces of equal dimension. Therefore, it is surjective as well. By
homogeneity it follows that ' is a submersion everywhere.
Theorem 17.4 (The isomorphism theorem for Lie groups). Let ' W G ! G 0 be a homomor-
phism of Lie groups. Then H WD ker ' is a closed normal subgroup of G: Moreover, the induced
homomorphism 'N W G=H ! G 0 is a smooth injective immersion. If ' is surjective, then 'N is an
isomorphism of Lie groups.
Proof: The following diagram is a commutative diagram of group homomorphisms
'
G ! G0
# %'N
G=H
Moreover, since is a submersion, whereas ' is smooth, it follows that 'N is smooth. Hence 'N is
an injective homomorphism of Lie groups. It follows by Lemma 7.6 that ' is an immersion. Now
assume that ' is surjective. Then 'N is surjective, and it follows by application of Lemma 17.2
that 'N is a submersion. We conclude that 'N is a local diffeomorphism, hence a diffeomorphism,
hence an isomorphism of Lie groups.
59
Example 17.5 The isomorphism of Proposition 10.3 is an isomorphism of Lie groups.
Example 17.6 Let G be a Lie group. Then Ad is a Lie group homomorphism from G into
GL.g/: It induces an injective Lie group homomorphism G= ker Ad ! GL.g/; realizing the
image Ad.G/ as a Lie subgroup of GL.g/: If G is connected, then ker Ad is the center Z.G/ of
G; see exercises. Consequently, Ad.G/ ' G=Z.G/ in this case.
Proposition 17.7 Let ' W G ! G 0 be a homomorphism of Lie groups. Then '.G/ is an open
subgroup of G 0 if and only if ' is submersive.
Proof: If ' is submersive, then '.G/ is open. Thus, it remains to prove the only if statement.
Let H be the kernel of ': Then by Theorem 17.4 it follows that the induced map 'N W G=H ! G 0
is an injective homomorphism of Lie groups. By application of Lemma 17.2 it follows that 'N is
a submersion. Since 'N D ' ; whereas is surjective, we now deduce that ' is a submersion
everywhere.
We end this section with a discussion of the Lie algebra of a quotient of a Lie group by a
closed normal subgroup.
Definition 17.8 Let l be a Lie algebra. An ideal of l is by definition a linear subspace a of l such
that l; a a; i.e. X; Y 2 a for all X 2 l; Y 2 a:
Lemma 17.10 (a) Let l be a Lie algebra, a l an ideal. Then the quotient (linear) space
l=a has a unique structure of Lie algebra such that the canonical projection W l ! l=a is a
homomorphism of Lie algebras.
(b) Let ' W l ! l0 be a homomorphism of Lie algebras, with kernel a: Then a is an ideal in l
and ' factors through an injective homomorphism of Lie algebras 'N W l=a ! l0 :
Lemma 17.11 Let G be a Lie group and let h be a subalgebra of its Lie algebra g:
(a) h is an ideal if and only if h is invariant under Ad.Ge /:
60
Lemma 17.12 Let H be a closed subgroup of the Lie group G: If H is normal, then its Lie
algebra h is an ideal in g: Moreover, the tangent map at e of the canonical projection W G !
G=H induces an isomorphism from the quotient Lie algebra g=h onto the Lie algebra of the Lie
group G=H:
Proof: Let l D T.e/ .G=H / be equipped with the Lie algebra structure induced by the Lie
group structure of G=H: The tangent map of the canonical projection W G ! G=H is a
Lie algebra homomorphism from g onto l: On the other hand, its kernel is h: Hence, by Lemma
17.10, factors through a Lie algebra isomorphism from g=h onto l:
Remark 17.13 Accordingly, if G is a Lie group and H a closed normal subgroup, then we shall
identify the Lie algebra of G=H with g=h via the isomorphism described above. In this fashion,
becomes the canonical projection g ! g=h:
Corollary 17.14 Let ' W G ! G 0 be a homomorphism of Lie groups, with kernel H:
(a) The induced map 'N W G=H ! G 0 is a homomorphism of Lie groups.
(b) Put ' D Te ': Then ker ' equals the Lie algebra h of H:
N is the linear map g=h ! g0 induced by ' :
(c) The tangent map 'N D TeN .'/
(d) If ' is surjective, then 'N and 'N are isomorphisms.
Proof: Assertion (a) follows by application of Theorem 17.4. Let W G ! G=H be the
canonical projection. Then is a homomorphism and a smooth submersion. By the preceding
remark, its tangent map is identified with the natural projection g ! g=h: From 'N D '
it follows by differentiation at e and application of the chain rule that 'N D ' : Since 'N is
a smooth immersion by Theorem 17.4, it follows that ker ' D ker D h: Hence, (b). From
'N D ' we also deduce (c). If ' is surjective, then 'N is an isomorphism of Lie groups by
Theorem 17.4. Hence, 'N is an isomorphism of Lie algebras.
61
Lemma 18.1 Let M be a locally compact Hausdorff space equipped with a right H -action by
continuous transformations. Then the following assertions are equivalent.
(a) The action of H on M is continuous, proper and free for the discrete topology on H:
(b) The action of H on M is properly discontinuous and the associated quotient space M=H
is Hausdorff.
Proof: Assume (a), and fix m 2 M: Then there exists a compact neighborhood N of m: The set
HN;N of h 2 H with N h \ N ; is compact in H; hence finite. Put C D HN;N n feg: For
every h 2 C we may select an open neighborhood Uh 3 m such that Uh h \ Uh D ; (observe
that mh m by freeness and use continuity of the action). Let U be the intersection of the finite
collection of open sets Uh .h 2 C / with the interior of N: Then U is open and U h \ U D ; for
all h 2 H n feg: It follows that the action of H is properly discontinuous. By the same argument
as in the proof of Theorem 13.5 it follows that CH is closed for every compact subset C M:
By Lemma 11.8 we conclude that G=H is Hausdorff.
Next assume (b), and let H be equipped with the discrete topology. We will first show that
the action map W M H ! M is continuous. Let U M be open. Then for each h 2 H the
set U h 1 is open in M: Hence U h 1 fhg is open in M H: The preimage 1 .U / equals the
union of these sets for h 2 H; hence is open.
Now suppose that C1 ; C2 are compact subsets of M: We will show that the set
HC1 ;C2 D fh 2 H j C1 h \ C2 ;g
62
In these notes the (complex linear) space of densities on V is denoted by DV: A density ! 2 DV
is called positive if it is non-zero and has values in 0; 1: The set of such densities is denoted
by DC V: It is obviously non-empty.
If ' is a linear isomorphism from V onto a real linear space W , then the map ' W ! 7! ! ' n
is a linear isomorphism DW ! DV of the associated spaces of densities. Indeed, if ! 2 DW;
T 2 End.V /; then
where dx denotes normalized Lebesgue measure. If ' is a diffeomorphism from U onto a second
open subset V Rn ; then, for g 2 Cc .V /; we have ' .g/.x/ D g.'.x// jdetD'.x/j .'.x//:
Thus, by the substitution of variables theorem:
Z Z
' !D ! .! 2 c .DT V //: (17)
U V
63
Let now .; / be a coordinate chart of X: If ! is a continuous density on X with compact
support supp ! ; then we define
Z Z
! WD . 1 / !:
X ./
This definition is unambiguous, because if .0 ; 0 / is a second chart such that supp ! 0 ; then
Z Z Z
0 1 0 1 0 1
. / ! D . / . / ! D . 1 / !
0 .0 / ./ ./
Just as in the theory of integration of differential forms one shows that this definition is inde-
pendent of the particular choice of partition of unity. Note that integration of forms is oriented,
whereas the present integration
R of densities is non-oriented.
We note that ! 7! X ! is a linear map c DTX ! C: Moreover, the following lemma is an
easy consequence of the definitions (reduction to charts et cetera).
We now turn to the situation that G is a Lie group acting smoothly from the left on a smooth
manifold M: If g 2 G; we write lg for the diffeomorphism M ! M; m 7! gm:
64
The following result will be very important for applications.
Lemma 19.5 Let ! be a G-invariant continuous density on M: Then for every f 2 Cc .M / and
all g 2 G we have: Z Z
lg .f / ! D f !: (18)
M M
Here lg f WD f lg :
Proof: We note that by invariance of ! we have lg .f /! D lg .f /lg .!/ D lg .f !/: Now observe
that lg is a diffeomorphism of M and apply the substitution of variables theorem (Proposition
19.3) with ' D lg :
Lemma 19.6 Let G be a Lie group and let .DT G/G denote the space of left invariant contin-
uous densities on G:
'
(a) The evaluation map W ! 7! !.e/ defines a linear isomorphism .DT G/G ! Dg:
(b) A density ! 2 .DT G/G is positive if and only if !.e/ is positive.
Proof: The map is linear. If ! is a left invariant continuous density on G; then !.g/ D
..lg 1 / !/.g/ D Te .lg / 1 !.e/ for all g 2 G: Hence, has trivial kernel. On the other hand, if
!0 is a density in Dg then the formula
1
!.g/ WD Te .lg / !0 (19)
defines a continuous density on G whose value at e is !0 : By application of the chain rule for
tangent maps it follows that this density is left G-invariant. Thus, (a) follows. Assertion (b)
follows from (19).
The following result is an immediate consequence of the above lemma.
Corollary 19.7 Every Lie group G has a left (resp. right) invariant positive density. Two such
densities differ by a positive factor.
R
If ! is a density on G; then the map Cc .G/ ! R; f 7! I.f / D G f ! is continuous linear,
hence aRRadon measure on G: For this reason we shall often write dx for an invariant density on
G; and G f .x/ dx for the associated invariant integral of a function f 2 Cc .G/: Note that in the
example of G D Rn with addition, dx is a (complex) multiple of Lebesgue measure. Positivity
then means that the multiple is positive, and invariance corresponds with translation invariance
of the Lebesge measure.
We now recollect some of the above results in the present notation. Let dx be a left invariant
positive density on G: (Analogous statements will be valid for right invariant positive densities.)
R
Proposition 19.8 The map f 7! I.f / D G f .x/ dx is a complex linear functional on Cc .G/:
It satisfies the following, for every f 2 Cc .G/:
65
(a) If f is real then so is I.f /I if f 0 then I.f / 0:
(b) If f 0 and I.f / D 0 then f D 0:
(c) For every y 2 G W Z Z
f .yx/ dx D f .x/ dx: (20)
G G
Proof: Assertion (a) follows from the positivity of !: Assertion (b) is immediate from Lemma
19.2. Finally (c) is a reformulation of Lemma 19.5.
Remark 19.9 One can show that up to a positive factor the linear functional I is uniquely
determined by the requirement I 0 and the properties (a) and (c). In particular property (b) is
a consequence. For details we refer the reader to the book by Brocker and tom Dieck.
It follows from the proposition that the Radon measure associated with a left invariant density
is left invariant, non-trivial and positive.
In the literature a left G-invariant positive Radon measure on G is called a left Haar mea-
sure of G: The above statement about the uniqueness of I is referred to as uniqueness of the
Haar measure. More generally a left (resp. right) Haar measure exists (and is unique up to a
positive factor) for any locally compact topological group. Of course one cannot use the present
differential geometric method of proof to establish the existence and uniqueness result in that
generality.
Lemma 19.10 Let G be a compact Lie group. Then there exists a unique left invariant density
dx on G with Z
dx D 1:
G
This density is positive.
Proof: Fix a positive Rdensity on G: Then it follows from assertions (a) and (b) of Proposition
19.8 for f D 1; that G equals a positive constant c > 0: The densitity dx D c 1 satifies
the above. This proves existence. If ! is a density with the same property, then ! D Cdx for a
constant C 2 C: Integration over G shows that C D 1: This establishes uniqueness.
Remark 19.11 The density of the above lemma is called the normalized left invariant density
of G: The associated Haar measure is called normalized Haar measure.
The following result expresses how left invariant densities behave under right translation.
Lemma 19.12 Let dx be a left invariant density on a Lie group G: Then for every g 2 G;
66
Proof: Without loss of generality we may assume that dx is non-zero. For g; h 2 G we have that
lh rg D rg lh ; hence rg lh D lh rg and we see that lh .rg .dx// D rg .lh dx/ D rg .dx/: It follows
that rg .dx/ is a left invariant positive density. This implies that rg .dx/ D c dx for a non-zero
constant c: Applying lg 1 to both sides of this equation we find Cg 1 dx D c dx: Evaluating
both sides of the latter identity in e we obtain
c dx.e/ D Te .Cg 1 / dx.e/ D Ad.g 1 / dx.e/ D jdetAd.g 1
/j dx.e/:
It follows that c D jdetAd.g/j 1 :
A Lie group G with jdetAd.g/j D 1 for all g 2 G is said to be unimodular. The following
result is an immediate consequence of the lemma.
Corollary 19.13 Let G be a unimodular Lie group. Then every left invariant density is also
right invariant.
Lemma 19.14 Let G be a compact Lie group. Then G is unimodular.
Remark 19.15 It follows that the normalized Haar measure of a compact Lie group is bi-
invariant.
Proof: The map x 7! jdetAd.x/j is a continuous group homomorphism from G into the group
.RC ; / of positive real numbers equipped with multiplication. Its image H is a compact sub-
group of .RC ; /: Now apply the lemma below to conclude that H D f1g:
67
The fiber of the bundle DT .G=H / over eH is identified with D.g=h/: Thus A.h/ is an
automorphism of the associated space of densities D.g=h/: Note that for ! 2 D.g=h/ we have:
jdet Ad.h/jg j
A.h/ ! D jdet A.h/j ! D !: (21)
jdet Ad.h/jh j
We write D.g=h/H for the linear space of densities ! on g=h satisfying A.h/ ! D !: Such
densities are called H -invariant. Since D.g=h/ is one dimensional, the space of H -invariant
densities is either 0 or 1 dimensional. In view of (21) the latter is the case if and only if
jdetAd.h/jg j D jdetAd.h/jh j for all h 2 H:
Lemma 19.18
(a) The evaluation map W ! 7! !.eH / defines a bijection from .DT .G=H //G onto
D.g=h/H : This bijection maps positive densities onto positive densities.
(b) The space of G-invariant densities on G=H is at most one dimensional. It is one dimen-
sional if and only if
Proof: Clearly is a linear map. Assume that ! is a G-invariant density on G=H: Then for
g 2 G we have: TeH .lg / !.gH / D lg .!/.eH / D .!/; hence
This shows that the map has a trivial kernel, hence is injective, and that its image is contained
in D.g=h/H : To establish its surjectivity, let !0 2 D.g=h/H : Then for all h 2 H we have
Note that the right hand side of this equation stays the same if g is replaced by gh; h 2 H: Hence
the definition is unambiguous. One readily verifies that ! thus defined is smooth, G-invariant,
and has image !0 under : This proves (a); the statement about positivity is obvious from the
above.
From (a) it follows that the dimension of .DT .G=H //G equals dimD.g=h/I hence, it is at
most one. The final assertion now follows from what was said in the preceding text.
Corollary 19.19 Let G be a Lie group, H a compact subgroup. Then G=H has a G-invariant
positive density. Two such densities differ by a positive factor.
68
Proof: For h 2 H; we put
jdetAd.h/jg j
.h/ D :
jdetAd.h/jh j
Clearly is a Lie group homomorphism from H to the group RC consisting of the positive real
numbers, equipped with multiplication. Thus, .H / is a compact subgroup of RC : In view of
Lemma 19.16 this implies that .H / D f1g: The result follows.
Example 19.20 As S n ' SO.n C 1/=SO.n/; see Example 15.6, it follows that S n has a unique
SO.nC1/-invariant density of total volume 1: Similarly, Pn .R/ has a unique SO.nC1/-invariant
density of total volume 1I see Example 15.7. Real projective space is non-orientable, so it does
not have a volume form, i.e., a nowhere vanishing exterior differential form of top degree. This
problem of possible non-orientability of homogeneous spaces has been our motivation in using
densities rather than forms to define invariant integration.
20 Representations
In this section G will always be a Lie group.
In the following we will give some of the basic definitions of representation theory with V a
complete locally convex space over C: Every Banach space is an example of such a space. Natu-
ral spaces of importance for analysis, like C.M /; Cc .M /; C 1 .M /; Cc1 .M /; with M a smooth
manifold, and also the spaces D 0 .M / and E 0 .M / of distributions and compactly supported distri-
butions, respectively, are complete locally convex, but in general not Banach. Of course Hilbert
spaces are Banach spaces; thus, they are covered as well.
Remark 20.2 If G is just a group, and V just a linear space, one defines a representation of G
in V similarly, but without the requirement of continuity.
Example 20.3
(a) Let G X ! X; .g; x/ 7! gx be a left action of G on a set X; and let F .X/ denote the
space of functions X ! C: Then the action naturally induces the representation L of G
on F .X/ given by
Lg '.x/ D '.g 1 x/;
for ' 2 F .X/; g 2 G and x 2 X:
69
(b) Let L be the action of G on F .G/ induced by the left action G G ! G; .g; x/ 7! gx:
This is called the left regular representation of G: It is given by the formula Lg '.x/ D
'.g 1 x/; for x; g 2 G:
Similarly, the right multiplication of G on itself induces the right regular representation
of G on F .G/ given by
Rg '.x/ D '.xg/;
for ' 2 F .G/; g; x 2 G: These representations leave the subspace C.G/ F .G/ in-
variant. Similarly, if dx is a left or right invariant Haar measure on G; then the associated
space L2 .G/ of square integrable functions is invariant under both L and R: One can show
that the restrictions of L and R to L2 .G/ are continuous, see Proposition 20.10.
(c) The natural action of SU.2/ on C2 induces a representation of SU.2/ on F .C2 / given
by
.x/'.z/ D '.x 1 z/ D '.z N 2 ; z1 C z2 /;
N 1 C z
for ' 2 F .C2 /; z 2 C2 and
N
xD 2 SU.2/;
N
Proof: By finite dimensionality of V; the group GL.V / is a Lie group. The map W x 7! .x/ is
a homomorphism from G to GL.V /: The hypothesis that the representation is continuous means
that the map .x; v/ 7! .x/v is continuous. By finite dimensionality of V this implies that
W G ! GL.V / is continuous. By Corollary 9.3 it follows that W G ! GL.V / is smooth.
This in turn implies that .g; v/ 7! .g/v is smooth G V ! V:
In the setting of the above lemma, the tangent map of W G ! GL.V / at e is a Lie algebra
homomorphism W g ! End.V /; where the latter space is equipped with the commutator
bracket. This motivates the following definition.
X; Y v D XY v YXv .X; Y 2 l; v 2 V /:
70
Remark 20.6 Similarly, a complete locally convex space V; equipped with a continuous repre-
sentation of a Lie group G; will sometimes be called a G-module
Proof: That (b) follows from (a) is obvious. We will establish the converse implication by
application of the Banach-Steinhaus (or uniform boundedness) theorem.
Assume (b). Fix x0 2 G: If v 2 V then .x/v D .x0 /.x0 1 x/vI using (b) we see that
x 7! .x/v is continuous at x0 :
Now fix v0 2 V: Select a compact neighborhood N of x0 in G: Then f.x/ j x 2 N g is a
collection of continuous linear maps V ! V: Moreover, for every v 2 V; the map x 7! k.x/vk
is continuous, hence bounded on N: By the uniform boundedness theorem it follows that the
collection of operator norms k.x/k; for x 2 N is bounded, say by a constant C > 0: It follows
that for x 2 N; v 2 V we have
The second term on the right-hand side tends to 0 if x ! x0 ; by (b). Hence .x; v/ 7! .x/v is
continuous in .x0 ; v0 /:
Remark 20.9 The above proof is based on the principle of uniform boundedness, and readily
generalizes to the category of complete locally convex spaces for which this principle holds, the
so called barrelled spaces.
Proposition 20.10 Let X be a manifold equipped with a continuous left G-action. Let dx be a
G-invariant positive continuous density on X: Then the natural representation L of G in L2 .X/
is continuous.
Proof: In view of the previous proposition it suffices to show that for every ' 2 L2 .X/ the
map W x 7! Lx '; G ! L2 .X/ is continuous at e: Thus we must estimate the L2 -norm of the
function Lx ' ' as x ! e: Let > 0: Then there exists a 2 Cc .X/ such that k' k2 < 13 :
71
Let g 2 Cc .G/ be a non-negative function such that g D 1 on an open neigbourhood of supp :
Then for x sufficiently close to e we have g D 1 on suppLx : Thus for such x we have:
2
kLx ' 'k2 C kLx k2
3
2
D C k.Lx /gk2
3
2
C kLx k1 kgk2 :
3
Fix a compact neighborhood N of supp : For x sufficiently close to e one has suppLx N:
By uniform continuity of on N; it now follows that kLx k1 kgk2 < 3 for x sufficiently
close to e:
Remark 20.12 Note that for a finite dimensional representation .; V / an invariant subspace is
automatically closed. Thus, such a representation is irreducible if the only invariant subspaces
are 0 and V:
Remark 20.14 Let V be a complex linear space. Then by a sesquilinear form on V we mean a
map W V V ! C which is linear in the first variable, and conjugate linear in the second, i.e.,
N
.v; w C w 0 / D .v; w/ C .v; w 0 / for all v; w; w 0 2 V; 2 C:
A Hermitian inner product on V is a sesquilinear form h ; i that is conjugate symmetric, i.e.
hv ; wi D hw ; vi; and positive definite, i.e., hv ; vi 0 and hv ; vi D 0 ) v D 0 for all v 2 V:
Finally, we recall that a complex Hilbert space is a complex linear space H equipped with a
Hermitian inner product h ; i; whose associated norm is complete.
Proposition 20.17 Let G be compact, and suppose that .; V / is a continuous finite dimen-
sional representation of G: Then is unitarizable.
72
Proof: Let dx denote right Haar measure on G; and fix any positive definite Hermitian inner
product h ; i1 on V: Then we define a new Hermitian pairing on V by
Z
hv ; wi D h.x/v ; .x/wi1 dx .v; w 2 V /:
G
Notice that the integrand v;w .x/ D h.x/v ; .x/wi1 in the above equation is a continuous
function of x: We claim that the pairing thus defined is positive definite. RIndeed, if v 2 V
then the function v;v is continuous and positive on G: Hence hv ; vi D G v;v .x/ dx 0
by positivity of the measure. Also, if hv ; vi D 0; then v;v 0 by Lemma 19.2, and hence
hv ; vi D v;v .e/ D 0; and positive definiteness follows.
Finally we claim that is unitary for the inner product thus defined. Indeed this follows from
the invariance of the measure. If y 2 G; and v; w 2 V; then
Z Z
h.y/v ; .y/wi D v;w .xy/dx D v;w .x/dx D hv ; wi:
G G
Proof: Let v 2 H2 and let x 2 G: We will show that .x/v 2 H2 : If w 2 H1 ; then .x 1 /w
belongs to H1 as well, so that h.x/v ; wi D hv ; .x 1 /wi D 0: It follows that .x/v 2 H1? :
Proof: Fix an inner product for which is unitary, and apply the above lemma repeatedly.
Corollary 20.20 Let .; V / be a continuous finite dimensional representation of a compact Lie
group. Then every invariant subspace of V has a complementary invariant subspace. Moreover,
admits a decomposition as a finite direct sum of irreducible representations.
Proof: By Proposition 20.17 is unitarizable. Now apply Lemma 20.18 and Corollary 20.19.
with v 2 V and 2 V :
73
Remark 20.22 Note that the map x 7! .x/ is smooth, so that every matrix coefficient belongs
to C 1 .G/:
If h ; i is a Hermitian inner product on V; then a matrix coefficient of may also be charac-
terized as a function of the form
Let now .; V / be a finite dimensional unitary representation of G; and fix an orthonormal
basis u1 ; : : : ; un of V: Then for every x 2 G we define the matrix M.x/ D Mu .x/ by
This is just the matrix of .x/ with respect to the basis u: Note that it is unitary. Note also
that M.xy/ D M.x/M.y/: Thus M is a continuous group homomorphism from G to the group
U.n/ of unitary n n matrices.
The representations 1 and 2 are said to be equivalent if there exists a topological linear iso-
morphism T from V1 onto V2 which is equivariant.
If the above representations are finite dimensional, then one does not need to require T to be
continuous, since every linear map V1 ! V2 has this property. In the case of finite dimensional
representations we shall write HomG .V1 ; V2 / for the linear space of interwining linear maps
V1 ! V2 and EndG .V1 / for the space of intertwining linear endomorphisms of V1 :
If V is a complex linear space, we write End.V / for the space of linear maps from V to itself,
and GL.V / for the group of invertible elements in End.V /: If is a representation of G in V ,
then we may define a representation Q of G in End.V / by
Q
.g/A D .g/A.g/ 1 :
74
Exercise 20.24 Let .j ; Vj /; for j D 1; 2; be two finite dimensional representations of G:
Show that 1 and 2 are equivalent if and only if there exist choices of bases for V1 and V2 ; such
that for the associated matrices one has:
mat1 .x/ D mat2 .x/:
Example 20.25 We recall that SU.2/ is the group of matrices of the form
N
gD
N
with ; 2 C and jj2 C jj2 D 1: The group SU.2/ acts on C2 in a natural way, and we have
the associated representation on the space P .C/ of polynomial functions p W C2 ! C: It is
given by the formula
.g/p.z/ D p.g 1 z/ D p.z N 2 ; z1 C z2 /
N 1 C z
The subspace Pn D Pn .C2 / of homogeneous polynomials of degree n is an invariant subspace
for : We write n for the restriction of to Pn :
We will now discuss a result that will allow us to show that the representations n of the
above example are irreducible. We first need the following lemma from linear algebra.
Lemma 20.26 Let V be a finite dimensional complex linear space, and let A; B 2 End.V / be
such that AB D BA: Then A leaves ker B; imB and all the eigenspaces of B invariant.
Proof: Elementary, and left to the reader.
From now on all representations of G are assumed to be continuous.
Lemma 20.27 (Schurs lemma) Let .; V / be a finite dimensional representation of G: Then
the following holds.
(a) If is irreducible then EndG .V / D C IV :
(b) Conversely, if is unitarizable and EndG .V / D C IV ; then is irreducible.
Proof: (a) Suppose that is irreducible, and let A 2 End.V /G : Let 2 C be an eigenvalue
of A; and let E D ker.A I/ be the associated eigenspace. Note that for non-triviality of this
eigenspace we need V to be complex. For every x 2 G we have that .x/ commutes with A;
hence leaves E invariant. In view of the irreducibility of it now follows that E D V; hence
A D I:
(b) By unitarizability of ; there exists a positive definite inner product h ; i for which
is unitary.
Let 0 W V be a G-invariant subspace. For the proof that is irreducible it suffices
to show that we must have W D V: Let P be the orthogonal projection V ! W: Since W
and W ? are both G-invariant, we have, for g 2 G; that .g/P D .g/ D P .g/ on W; and
.g/P D 0 D P .g/ on W ? : Hence P 2 EndG .V /; and it follows that P D I for some
2 C: Now P 0; hence 0: Also, P 2 D P; hence 2 D ; and we see that D 1:
Therefore P D I; and W D V:
75
We will now apply the above lemma to prove the following.
Proposition 20.28 The representations .n ; Pn.C2 // of SU.2/; for n 2 N; are irreducible.
For the proof we will need compactness of SU.2/: In fact we have the following more general
result.
Exercise 20.29 For n 1; let M.n; R/ and M.n; C/ denote the linear spaces of n n matrices
with entries in R and C respectively. Show that SU.n/ is a closed and bounded subset of M.n; C/:
Show that SO.n/ D SU.n/ \ M.n; R/: Finally show that the Lie groups SO.n/ and SU.n/ are
compact.
Proof of Proposition 20.28: Let n 0 be fixed, and put D n and V D Pn .C2 /: Then n is
unitarizable, since SU.2/ is compact. Suppose that A 2 End.V / is equivariant. Then in view of
Lemma 20.27 (b) it suffices to show that A is a scalar.
For 0 k n we define the polynomial pk 2 V by
pk .z/ D z1n k k
z2 :
Then fpk j 0 k ng is a basis for V: For ' 2 R we put
i'
e 0 cos ' sin '
t' D i' ; r' D :
0 e sin ' cos '
Then
T D ft' j ' 2 Rg and R D fr' j ' 2 Rg
are (closed) subgroups of SU.2/: One readily verifies that for 0 k n and ' 2 R we have:
.t' /pk D e i.2k n/'
pk :
Thus every pk is a joint eigenvector for T: Fix a ' such that the numbers e i.2k n/' are mutually
different. Then for every 0 k n the space Cpk is eigenspace for .t' / with eigenvalue
e i.2k n/' : Since A and .t' / commute it follows that A leaves all the spaces Cpk invariant.
Hence there exist k 2 C such that
Apk D k pk ; 0 k n:
Let E0 be the eigenspace of A with eigenvalue 0 : We will show that E0 D V; thereby complet-
ing the proof. The space E0 is SU.2/-invariant, and contains p0 : Hence it contains .r' /p0 for
every ' 2 R: By a straightforward computation one sees that
n
!
X n
.r' /p0 .z1 ; z2 / D .cos ' z1 C sin ' z2 /n D cosn k ' sink ' pk :
k
kD0
From this one sees by application of A and using the intertwining property, that
n
!
X n
cosn k ' sink ' .0 k / pk D 0;
k
kD0
for all ' 2 R: By linear independence of the pk ; it follows that k D 0 ; for every 0 k n:
Hence E0 D V:
76
We end this section with two useful consequences of Schurs lemma.
Proof: Let T be intertwining, and non-trivial. Then ker T V is a proper G-invariant subspace.
Hence ker T D 0; and it follows that T is injective. Therefore its image imT is a non-trivial G-
invariant subspace of V 0 : It follows that imT D V 0 ; hence T is a bijection, contradicting the
inequivalence.
If .; V / is a representation for a group G; then a sesquilinear form on V is called equiv-
ariant if ..g/v; .g/w/ D .v; w/ for all v; w 2 V; g 2 G:
Lemma 20.31 Let .; V / be an irreducible finite dimensional unitary representation of a lo-
cally compact group G: Then the equivariant sesquilinear forms on V are precisely the maps
W V V ! C of the form D h ; i; 2 C: Here h ; i denotes the (equivariant) inner
product of the Hilbert space V:
21 Schur orthogonality
Assumption In the rest of these notes every finite dimensional representation of a Lie group will
be assumed to be continuous, unless specified otherwise.
In this section G will be a compact
R Lie group, unless stated otherwise. Let dx be the unique
left invariant density on G with G dx D 1I for its existence, see Lemma 19.10. Then dx is
positive. By Remark 19.15, the density dx is right invariant as well.
If is a finite dimensional irreducible unitary representation of G we write
C.G/ (22)
for the linear span of the space of matrix coefficients of : Notice that the space C.G/ does not
depend on the chosen (unitary) inner product on V: Thus, by Proposition 20.17 we may define
C.G/ for any irreducible finite dimensional (continous) representation of G:
There is a nice way to express sums of matrix coefficients of a finite dimensional unitary
representation .; V / of G by means of the trace of a linear map. Let v; w 2 V: Then we shall
write Lv;w for the linear map V ! V given by
77
One readily sees that
tr.Lv;w / D hv ; wi; v; w 2 V: (23)
Indeed both sides of the above equation are sesquilinear forms in .v; w/; so it suffices to check
the equation for v; w members of an orthonormal basis, which is easily done.
It follows from the above equation that
Hence every sum m of matrix coefficients is of the form m.x/ D tr..x/A/; with A 2 End.V /:
Conversely if fek j 1 k ng is an orthonormal basis for V; then one readily sees that any
endomorphism A 2 End.V / may be expressed as
X
AD hAej ; ei iLei ;ej :
1i;j n
Using this one may express every function of the form x 7! tr..x/A/ as a sum of matrix
coefficients.
We now define the linear map T W End.V / ! C.G/ by
T .A/.x/ D tr..x/A/; x 2 G;
for every A 2 End.V /: Let be irreducible, then it follows from the above discussion that T
maps V onto C.G/ : Define the representation of G G on End.V / by
.R L/.x; y/ WD Rx Ly D Ly Rx :
Lemma 21.1 Let .; V / be a finite dimensional irreducible representation of G: Then C.G/
is invariant under R L: The map T W V ! C.G/ is surjective, and intertwines the represen-
tations and R L of G G:
Proof: We first prove the equivariance of T W End.V / ! C.G/: Let A 2 End.V / and x; y 2 G;
then for all g 2 G;
Note that it follows from this equivariance that the image of T is R L-invariant. In an earlier
discussion we showed already that im.T / D C.G/ :
78
Proof: Let V; V 0 be the associated representation spaces. Then by equivalence there exists a
linear isomorphism T W V ! V 0 such that T .x/ D 0 .x/ T for all x 2 G: Hence for
A 2 End.V / and x 2 G;
1
T 0 .TAT /.x/ D tr. 0 .x/TAT 1
/ D tr.T 1
0 .x/TA/ D tr..x/A/ D T .A/.x/:
Now use that T and T 0 have images C.G/ and C.G/ 0 ; respectively, by Lemma 21.1.
We now have the following.
Theorem 21.3 (Schur orthogonality). Let .; V / and . 0 ; V 0 / be two irreducible finite dimen-
sional representations of G: Then the following holds.
(a) If and 0 are not equivalent, then C.G/ ? C.G/ 0 (with respect to the Hilbert structure
of L2 .G/).
(b) Let V be equipped with an inner product for which is unitary. If v; w; v 0; w 0 2 V; then
the L2 -inner product of the matrix coefficients mv;w and mv 0 ;w 0 is given by:
Z
mv;w .x/ mv 0 ;w 0 .x/ dx D dim./ 1 hv ; v 0 ihw ; w 0 i (24)
G
Remark 21.4 The relations (24) are known as the Schur orthogonality relations. Of course the
assumption that dx is normalized is a necessary assumption for (24) to hold.
Proof: For w 2 V and w 0 2 V 0 we define the linear map Lw 0 ;w W V ! V 0 by Lw 0 ;w u D
hu ; wiw 0 : Consider the following linear map V ! V 0 ; defined by averaging,
Z
Iw 0 ;w D 0 .x/ 1 Lw 0 ;w .x/ dx:
G
Hence
Iw 0 ;w D .w 0 ; w/ IV D d 1 hw 0 ; wi IV :
Now apply (25) to prove (b).
79
Another way to formulate the orthogonality relations is the following (V is assumed to be
equipped with an inner product for which is unitary). If A 2 End.V /, let A denote the
Hermitian adjoint of A: Then one readily verifies that
.A; B/ 7! hA ; Bi WD trB A
Lv 0 ;w 0 D Lw 0 ;v 0 :
Lw 0 ;v 0 Lv;w D hv ; v 0 iLw 0 ;w :
hence
h T .Lv;w / ; T .Lv 0 ;w 0 /iL2 D d hmv;w ; mv 0 ;w 0 iL2 : (27)
From (26) and (27) we see that the Schur orthogonality relations may be reformulated as
for all v; v 0 ; w; w 0 2 V: The maps Lv;w ; for v; w 2 V; span the space End.V /: Hence the Schur
orthogonality relations are equivalent to the assertion that T is an isometry from End.V / into
C.G/ : We proved already that T is surjective onto C.G/ I hence T is a unitary isomor-
phism. The equivariance of T has been established before.
80
Remark 21.7 Since the representation is continuous, it is also smooth, hence 2 C 1 .G/:
Note that is a sum of matrix coefficients of : Thus, if G is compact and irreducible, then
2 C.G/ :
Proof: We equip V with an inner product for which is unitary, and define the associated inner
product on End.V / as above. Let ' 2 C.G/ : Then ' D T .A/ for a unique A 2 End.V /:
By equivariance of T ; the function ' is conjugation invariant if and only if A is G-intertwining,
which in turn is equivalent to A D cIV for a constant c 2 C: We observe that c D '.e/=d3=2 :
p a unique conjugation invariant function ' with '.e/ D d : For this
This implies that there exists
function we have c D 1= d and
p
'.x/ D T .cIV /.x/ D d tr..x/cIV / D tr.x/ D .x/:
22 Characters
In this section we assume that G is a Lie group. We shall discuss properties of characters of finite
dimensional representations of G:
If V is a finite dimensional complex linear space, we write End.V / for the space of complex
linear maps from V to itself, and det D detV and tr D trV for the complex determinant and trace
functions End.V / ! C:
81
The character of a finite dimensional representation .; V / of G is defined as in Defini-
tion 21.6.
Lemma 22.3 Let .; V / be a finite dimensional representation of G: Then, for all x; y 2 G;
1
.xyx / D .y/:
_ .x/ D .x 1
/ W v 7! v .x 1
/; .x 2 G/:
Proof: Let v1 ; : : : ; vn be a basis for V and let v 1 ; : : : ; v n be the dual basis for V ; i.e., v i .vj / D
ij : Then, for x 2 G; the matrix of _ .x/ with respect to the basis v1 ; : : : ; vn is given by
_ .x/ij D h.x 1 j
/ v ; vi i D hv j ; .x 1
/vi i D .x 1
/j i :
If is continuous, then its matrix coefficients are continuous functions. Therefore, so are the
matrix coefficients of _ ; and (a) follows. Assertion (b) follows from the above identity as well.
Characters of unitarizable representations have the following special property.
82
If .1 ; V1 / and .2 ; V2 / are two continuous representations of G; then we define the direct
sum representation D 1 2 in the direct sum V D V1 V2 by
for all v1 2 V1 ; v2 2 V2 :
Lemma 22.8 Let .1 ; V1 / and .2 ; V2 / be finite dimensional representations of G: Then the
character of their tensor product 1 2 is given by
Proof: Exercise for the reader. Establish, more generally, an identity of the form tr.A B/ D
tr.A/tr.B/; by choosing suitable bases.
Exercise 22.9 Recall the definition, for n 2 N; of the representation n of SU.2/ in the finite
dimensional space Pn .C2 / of homogeneous polynomial functions C2 ! C of degree n: Show
that the character n of n is completely determined by its restriction to T D ft' j ' 2 Rg: Hint:
use that every matrix in SU.2/ is conjugate to a matrix of T:
Show that:
sin.n C 1/'
n.t' / D ;
sin '
for ' 2 R: Here t' denotes the diagonal matrix with entries e i' and e i'
:
Assumption: In the rest of this section we assume that the Lie group G is compact. We denote
by h ; i the L2 -inner product with respect to the normalized Haar measure dx on G:
Lemma 22.10 Let ; be finite dimensional irreducible representations of G:
(a) If then h ; i D 1:
(b) If 6 then h ; i D 0:
Proof: This follows easily from Theorem 21.3.
83
Let be a finite dimensional representation of the compact group G: Then is unitarizable,
and therefore
Pn equivalent to a direct sum niD1 i of irreducible representations. It follows that
D iD1 i : Using the lemma above we see that for every irreducible representation of G
we have
#fi j i g D h ; i: (29)
In particular this number is independent of the particular decomposition of into irreducibles.
For obvious reasons the number (29) is called the multiplicity of in : We shall also denote it
by m.; /:
b denote the set of equivalence classes of finite dimensional irreducible representations
Let G
of G: Then by abuse of language we shall write 2 G b to indicate that is a representative for an
b (A better notation would perhaps be 2 G:/
element of G: b If 2 G b and m 2 N; then we write
m for (the equivalence class of) the direct sum of m copies of :
We have proved the folllowing lemma.
Lemma 22.11 Let be a finite dimensional representation of the compact group G: Then
M
m.; /;
b
2G
84
23 The Peter-Weyl theorem
In this section we assume that G is a compact Lie group. We denote by G b the set of (equivalence
classes of) irreducible continuous finite dimensional representations of G:
Definition 23.1 We define the space R.G/ of representative functions to be the space of func-
b
tions f W G ! C that may be written as a finite sum of functions f 2 C.G/ ; for 2 G:
Note that the space R.G/ is contained in C 1 .G/: Moreover, it is invariant under both the
left- and the right regular representations of G:
Exercise 23.2 Show that R.G/ is the linear span of the set of all matrix coefficients of finite
dimensional continuous representations of G: Hint: consider the decomposition of finite dimen-
sional representations into irreducibles.
Proposition 23.3 The space of representative functions decomposes according to the algebraic
direct sum M
R.G/ D C.G/ :
b
2G
The summands are mutually orthogonal with respect to the L2 -inner product. Every summand
C.G/ is invariant under the representation R L of G G: Moreover, the restriction of R L
to that summand is an irreducible representation of G G:
Proof: The orthogonality of the summands follows from Schur orthogonality. It follows that the
above sum is direct.
The map T W End.V / ! C.G/ is bijective and intertwines with R L: Hence it
suffices to show that is an irreducible representation of G G:
By a straightforward computation one checks that
for .x; y/ 2 G G: If dx and dy are normalized right Haar measure on G; then the product
measure dx dy is the normalized right Haar measure on G G: Moreover, by Fubinis theorem,
Z
2
k kL2 .GG/ D j .x/j2 j .y/j2 dx dy
ZGGZ
2 2
D j .x/j j .y/j dx dy
G G
D k kL2 .G/ k k2L2 .G/
2
D 1;
85
The proof of the following result is based on the spectral theorem for compact self-adjoint
operators in a Hilbert space. It will be given in the next section.
Let H be a collection of Hilbert spaces, indexed by a set A: Then the algebraic direct sum
M
H
2A
P P
Pa pre-Hilbert space when equipped with the direct sum inner product: h v ; w i D
is
hv ; w i: Its completion is called the Hilbert direct sum of the spaces H ; and denoted by
^
M
H : (30)
2A
This completion may be realized as the space of sequences v D .v /2A with v 2 H and
X
kvk2 D kv k2 < 1:
2A
Theorem 23.5 (The Peter-Weyl Theorem). The space L2 .G/ decomposes as the Hilbert sum
^
M
2
L .G/ D C.G/ ;
b
2G
each of the summands being an irreducible invariant subspace for the representation R L of
G G:
Exercise 23.6 Fix, for every (equivalence class of an) ireducible unitary representation .; V /
an orthonormal basis e;1 ; : : : ; e;dim./ : Denote the matrix coefficient associated to e;i and e;j
by m;ij : Use Schur orthogonality and the Peter-Weyl theorem to show that the functions
p
dim./ m;ij b 1 i; j dim./
2 G;
86
24 Appendix: compact self-adjoint operators
Definition 24.1 Let V; W be Banach spaces. A linear map T W V ! W is said to be compact if
the image T .B/ of the unit ball B D B.0I 1/ V has compact closure in W:
Lemma 24.2 Let V; W be Banach spaces, and let L.V; W / be the Banach space of bounded
linear operators V ! W; equipped with the operator norm. Then the subspace of compact
linear operators V ! W is closed in L.V; W /:
Remark 24.3 A linear map T W V ! W is said to be of finite rank if its image T .V / is finite
dimensional. Clearly an operator of finite rank is compact. Thus, if Tj is a sequence of operators
in L.V; W / all of which are of finite rank, and if Tj ! T with respect to the operator norm, then
it follows from the above result that T is compact.
We recall that a bounded linear operator T from a complex Hilbert space H to itself is said
to self-adjoint if T D T; or, equivalently, if hT v ; wi D hv ; T wi for all v; w 2 H:
We now recall the important spectral theorem for compact self-adjoint operators in Hilbert
space. It will play a crucial role in the proof of the Peter-Weyl theorem in the next section. For a
proof of the spectral theorem, we refer to a standard text book on functional analysis.
Theorem 24.4 Let T be a compact self-adjoint operator in the (complex) Hilbert space H: Then
there exists a discrete subset R n f0g such that the following hold.
(a) For every 2 the associated eigenspace H of T in H is finite dimensional.
(b) If ; 2 ; then H ? H :
(c) For every 2 ; let P denote the orthogonal projection H ! H : Then
X
T D P ;
2
We will end this section by describing a nice class of compact self-adjoint operators in
L2 .G/; for G a compact Lie group. First we examine the space of compactly supported con-
tinuous functions on product space.
87
Let X; Y be locally compact topological Hausdorff spaces. If ' 2 C.X/; and 2 C.Y /;
then we write ' for the continuous function on X Y defined by:
' W .x; y/ 7! '.x/ .y/:
The linear span of such functions in C.X Y / is denoted by C.X/ C.Y /: If ' 2 Cc .X/
and 2 Cc .Y / then ' is compactly supported. Hence the span Cc .X/ Cc .Y / of such
functions is a subspace of Cc .X Y /:
Proposition 24.5 Let X; Y be locally compact Hausdorff spaces. Then for every open subset
U X Y with compact closure, every 2 Cc .U / and every > 0; there exists a function
' 2 Cc .X/ Cc .Y / with supp' U and supz2U j.z/ '.z/j < : In particular, the space
Cc .X/ Cc .Y / is dense in Cc .X Y /:
Proof: Using Cc -partitions of unity for X and Y; we see that we may reduce to the case that
U D UX UY ; with UX and UY open neighborhoods with compact closures in X and Y
respectively.
Fix 2 Cc .X Y /; with K D supp U: Then, by compactness, K KX KY for
compact subsets KX UX and KY UY . Let > 0: Then by compactness there exists a finite
open covering fVj g of KX such that for every j and all x1 ; x2 2 Vj ; y 2 KY one has
.x1 ; y/ .x2 ; y/ < :
Without loss of generality we may assume that Vj UX for all j: Select a partition of unity f'j g
which is subordinate to the covering fVj g; and fix for every j a point j 2 Vj : Let x 2 KX ; y 2
KY : If j is such that x 2 Vj ; then j.xj ; y/ .x; y/j < : It follows from this that
X X
j 'j .x/.xj ; y/ .x; y/j D j 'j .x/.xj ; y/ 'j .x/.x; y/j
j j
X
'j .x/j.xj ; y/ .x; y/j
j
X
< 'j .x/ D :
j
Moreover, supp'j j UX UY UX KY U:
Let now G be a Lie group. We fix a left invariant density dx on G and equip G G with
the left invariant product of dx with itself. This product density, denoted dxdy; is determined
by the formula
Z Z Z Z Z
f .x; y/ dxdy D f .x; y/dx dy D f .x; y/dy dx;
GG G G G G
88
for f 2 Cc .X Y /:
If K 2 Cc .G G/; then we define the linear operator TK W Cc .G/ ! Cc .G/ by
Z
TK .'/.x/ D K.x; y/'.y/dy:
G
Lemma 24.6 Let K 2 Cc .G G/: Then the operator TK extends uniquely to a bounded
linear endomorphism of L2 .G/ with operator norm kTK kop kKk2 : Moreover, this extension
is compact.
Hence kTK 'k2 kKk2 k'k2 : This implies the first assertion, since Cc .G/ is dense in L2 .G/:
For the second assertion, note that by Proposition 24.5 there exists a sequence Kj in Cc .G/
Cc .G/ which converges to K with respect to the L2 -norm on G G: It follows that
Every operator TKj has a finite dimensional image hence is compact. The subspace of compact
endomorphisms of L2 .G/ is closed for the operator norm, by Lemma 24.2. Therefore, TK is
compact.
Let G be a Lie group, equipped with a left invariant density dx: If .; V / is a continuous
finite dimensional representation of G, then for f 2 Cc .G/ we define the linear operator .f / W
V ! V by Z
.f /v D f .x/.x/v dx:
G
Referring to integration with values in a Banach space, this definition actually makes sense if
is a continuous representation in a Banach space; it is readily seen that then .f / is a continuous
linear operator. In particular, the definition may be applied to the regular representations L and
R of G in L2 .G/: Thus, for f 2 Cc .G/ and ' 2 L2 .G/;
Z Z
R.f /'.x/ D f .y/'.xy/ dy D f .x 1 y/'.y/ dy .x 2 G/: (31)
G G
Of course, this formula can also be used as the defining formula, without reference to Banach-
valued integration.
89
Corollary 24.7 Assume that G is compact, and let f 2 C.G/: Then the operator R.f / W
L2 .G/ ! L2 .G/ is compact.
1
Proof: If ' 2 C.G/; then from (31) we see that R.f / D TK ; with K.x; y/ D f .x y/: The
result now follows by application of Lemma 24.6.
Remark 24.8 Note that for this argument it is crucial that G is compact. For if not, and f 2
Cc .G/; then the associated integral kernel K need not be compactly supported.
The following lemmas will in particular be needed for the right regular representation R:
Lemma 24.9 Let .; H/ be a unitary representation of G in a Hilbert space. Let f 2 Cc .G/;
then
.f / D .f /;
where f .x/ D f .x 1 /:
Corollary 24.11 Assume that G is compact, and let f 2 C.G/ be such that f D f: Then
R.f / (and L.f / as well) is a compact self-adjoint operator. If, in addition, f is conjugation
invariant then R.f / is G-equivariant.
Proof: This follows by combining Corollary 24.11 and Lemmas 24.9 and 24.10.
Lemma 25.1 Let ' 2 Cc .G/: Then R.'/ maps L2 .G/ into C.G/:
90
Proof: Let x0 2 G and let > 0: Since ' has compact support C WD supp'; it follows by the
principle of uniform continuity that there exists a compact neighborhood U of e in G such that
j'.u/ '.v/j < .2k1C k2 C 1/ 1 for all u; v 2 G with vu 1 2 U:
Let now f 2 L2 .G/: For x; y 2 G with x 2 x0 U we have .x0 1 y/.x 1 y/ 1 D x0 1 x 2 U;
hence
Z
jR.'/f .x/ R.'/f .x0 /j D j '.x 1 y/ '.x0 1 y/f .y/ dyj
Z G Z
jf .y/j dy D 1xC [x0 C jf .y/j dy
xC [x0 C G
k1xC [yC k2 kf k2 2 k1C k2 kf k2 kf k2 :
Lemma 25.2 Let f 2 L2 .G/ and let > 0: There exists an open neighborhood U of e in G
such that for all x 2 U we have kRx f f k2 <R : Moreover, if U is any neighborhood with this
property and if ' 2 Cc .U / satisfies ' 0 and G '.x/ dx D 1; then
Proof: The first assertion follows from the continuity of the map x 7! Rx f; see Proposition
20.10. Let U; ' be as stated. Then, for all x 2 G;
Z
R.'/f .x/ f .x/ D '.y/f .xy/ f .x/ dy:
G
Lemma 25.3 Let V be a finite dimensional right G-invariant subspace of L2 .G/: Then V
R.G/:
91
Proof: Decomposing V into a direct sum of irreducible subspaces, we see that we may reduce
the case that V is irreducible. We claim that V consists of continuous functions. For this we
observe that C.G/ \ V is an invariant subspace. Hence it suffices to show that V contains a
non-trivial continuous function. Fix f 2 V n f0g and fix 0 < < 1=2kf k2 : Choose U and '
as in Lemma 25.2. Then kR.'/f k > 1=2; hence R.'/f 0: From Lemma 25.1 it follows
that R.'/f 2 C.G/: Moreover, since V is right invariant, it follows that R.'/f 2 V: This
establishes the claim that V C.G/:
Choose an orthonormal basis . i / of V: Then for f 2 V we have
X
Rx f D hRx f ; i i i ;
i
hence by evaluation in e; X
f .x/ D hRx f ; ii i .e/:
i
Lemma 25.4 Let U be an open neighborhood of e in G: Then there exists a ' 2 Cc .U / such
that:
R
(a) ' 0 and G '.x/ dx D 1I
(b) ' D 'I
(c) ' is conjugation invariant.
Proof: From the continuity of the map x 7! x 1 one sees that there exists a compact neighbor-
hood V of e such that V U and V 1 U: For every x 2 G there exist an open neighborhood
Nx of x and a compact neighborhood Vx of e in V such that zyz 1 2 V for all z 2 Nx ; y 2 Vx :
By compactness of G finitely many of the Nx cover G: Let be the intersection of the corre-
sponding Vx : Then is a compact neighborhood of e and for all x 2 G and y 2 we have
xyx 1 2 V: R
Now select 0 2 Cc ./ such that 0 0 and G 0 .x/ dx D 1: Define
Z
1
.x/ D 0 .yxy / dy:
G
Corollary 25.5 Let f 2 L2 .G/; f 0: Then there exists a left and right G-equivariant
bounded linear operator T W L2 .G/ ! L2 .G/ with:
92
(a) Tf 0:
(b) T is self-adjoint and compact;
(c) T maps every right G-invariant closed subspace of L2 .G/ into itself.
Proof: Let D 12 kf k2 ; and fix an open neighborhood U of e in G that satisfies the assertion
of Lemma 25.2 Let ' 2 Cc .U / be as in Lemma 25.4, and define T D R.'/: Then kTf
f k < ; hence (a). Moreover, every closed right invariant subspace V of L2 .G/ equipped
with the restriction of R is a continuous representation in a Banach space, hence invariant under
T D R.'/: This implies (c).
The operator T is left G-equivariant, since L and R commute. It is right G-equivariant
because ' is conjugation invariant, cf. Lemma 24.10. Finally (b) follows from Corollary 24.11.
Proof of Propostion 23.4. The space R.G/ is left and right G-invariant, and by unitarity so is its
orthocomplement V: Suppose that V contains a non-trivial element f: Let T be as in Corollary
25.5. Then T jV W V ! V is a non-trivial compact self-adjoint operator which is both left and
right G-equivariant. By the spectral theorem for compact self-adjoint operators, Theorem 24.4,
there exists a 2 R, 0; such that the eigenspace V D ker.T IV / is non-trivial. By
compactness of T the eigenspace V is finite dimensional, and by equivariance of T it is both left
and right G-invariant. By Lemma 25.3 it now follows that V R.G/; contradiction. Therefore,
V must be trivial.
26 Class functions
By a class function on a compact Lie group G we mean a function f W G ! C that is conjugation
invariant, i.e., Lx Rx f D f for all x 2 G: The name class function comes from the fact that a
conjugation invariant function is constant on the conjugacy classes, hence may be viewed as a
function on the set of conjugacy classes.
The space C.G; class/ of continuous class functions is a closed subspace of C.G/ (with
respect to the sup norm). Its closure in L2 .G/ equals L2 .G; class/; the space of square integrable
class functions on G:
b we denote the orthogonal projection from L2 .G/ onto the finite dimensional sub-
If 2 G;
space C.G/ by
P W L2 .G/ ! C.G/
Note that P is equivariant for both the representations R and L of G: In particular, this implies
that P maps C.G; class/ into its intersection with C.G/ : Hence, by Lemma 21.8
It follows from this that the space R.G; class/ D C.G; class/ \ R.G/ of representative class
b
functions is the linear span of the characters ; 2 G:
93
b form a complete
Lemma 26.1 Let G be a compact Lie group. Then the characters ; 2 G;
2
orthonormal system for L .G; class/:
Proof: By Schur orthogonality, the characters form an orthonormal system. To establish its
b
completeness, let f 2 L2 .G; class/ and assume that f ? for all 2 G:
From P f 2 C.G/ D C ; we see that
P f D hP f ; i D hf ; i D 0:
Proof: By compactness, there exists a constant r > 0 such that r 1 < jzj < r for all z 2 H:
Let w 2 H; then applying the estimate to z D w n we obtain that r 1=n jwj r 1=n : Taking
the limit for n ! 1 we see that jwj D 1:
Proof: If x 2 G; then .y/.x/ D .yx/ D .xy/ D .x/.y/ for all y 2 G; hence .x/ is
equivariant, and it follows that
.x/ D .x/I; (33)
for some .x/ 2 C; by Schurs lemma. It follows from this that every linear subspace of V is
invariant. By irreducibility of this implies that the dimension of V must be one. From the fact
that is a representation it follows immediately that x 7! .x/ is a character. Applying the trace
94
to (33) we see that D ; the character of : Thus 7! induces a map from the space G b of
equivalence classes of finite dimensional irreducible representations to the set of multiplicative
characters of G: This map is injective by Corollary 22.13. If is a multiplicative character then
(33) defines an irreducible representation of G in C; and D : Therefore the map ! is
surjective onto the set of multiplicative characters.
Corollary 27.3 Assume that G is a commutative compact Lie group. Then the set of multiplica-
b is a complete orthonormal system for L2 .G/:
tive characters ; 2 G;
Proof: This follows immediately from the previous lemma combined with the theorem of Peter
and Weyl (Theorem 23.5).
In the present setting we define the Fourier transform fO W G
b ! C of a function f 2 L2 .G/
by
fO./ D hf ; i:
Let Gb be equipped with the counting measure. Then the associated L2 -space is l 2 .G/;
b the space
P
b ! C such that
of functions ' W G 2
b j'./j < 1; equipped with the inner product:
2G
X
h' ; i WD './ ./:
b
2G
Corollary 27.4 (The Plancherel theorem). Let G be a commutative compact Lie group. Then
the Fourier transform f 7! fO is an isometry from L2 .G/ onto l 2 .G/:
b Moreover, if f 2 L2 .G/;
then X
f D fO./ ;
b
2G
m W x 7! e i.mx/
95
Show that for f 2 L2 .G/; m 2 Zn we have:
Z 2 Z 2
O 1 i.m1 x1 CCmn xn /
f .m/ D ::: f .x1 ; : : : ; xn / e dx1 : : : dxn :
.2/n 0 0
in the L2 -sense.
96
Corollary 28.3 The linear span of the characters n; for n 2 N; is dense in C.G; class/:
where k k1 denotes the sup norm. Using this estimate we see that the linear span of the
characters n is dense in C.G; class/ with respect to the L2 -norm. Thus, if f 2 C.G; class/ is
perpendicular to all n ; then it follows that f ? C.G; class/: In particular, kf k22 D hf ; f i D 0;
which implies that f D 0:
Proof: Suppose not. Then there exists a 2 G b such that is not equivalent to n ; for every
n 2 N: Hence the class function is perpendicular to n for every n 2 N: This implies that
D 0: This is impossible, since .e/ D dim./ 1:
From the fact that every element of SU.2/ is conjugate to an element of T one might expect
that there should exist a Jacobian J W T ! 0; 1 such that for every continuous class function
f on SU.2/ we have Z Z 2
f .x/ dx D f .t' / J.t' / d':
SU.2/ 0
97
Proof: Consider the linear map L which assigns to f 2 C.G; class/ the expression on the left-
hand side minus the expression on the right-hand side of the above equation. Then we must show
that L is zero.
Obviously the linear functional L W C.G; class/ ! C is continuous with respect to the sup
norm. Hence by density of the span of the characters it suffices to show that L.n / D 0 for every
n 2 N: The function 0 is identically one; therefore left- and right-hand side of (35) both equal
1 if one substitutes f D 0 : Hence L.0 / D 0: On the other hand, if n 1; and f D n ; then
the left hand side of (35) equals hn ; 0 i D 0: The right hand side of (35) also equals 0; hence
L.n / D 0 for all n:
Remark 28.9 The interpretation of the above formula is that the integration over G D SU.2/
may be split into an integration over conjugacy classes, followed by an integration over the circle
group T:
Proof: Put Z
1
F .y/ D f .xyx / dx:
G
Then by bi-invariance of the Haar measure, F is a continuous class function. Hence by the
previous result
Z Z 2 Z
sin2 '
F .y/ dy D f .xt' x 1 / dx d':
G 0 G
On the other hand,
Z Z Z
F .y/ dy D f .xyx 1 / dx dy
G
ZG ZG
D f .xyx 1 / dy dx;
G G
by Fubinis theorem. By bi-invariance of the Haar measure, the inner integral is independent of
x: Therefore,
Z Z Z
F .y/ dy D f .y/ dy dx
G Z G G
D f .y/ dy:
G
98
We end this section with a description of all irreducible representations of SO.3/: From Sec-
tion 10 we recall that there exists a surjective Lie group homomorphism ' W SU.2/ ! SO.3/
with kernel ker ' D f I; I g: Accordingly, SO.3/ ' SU.2/=fI g (Thm. 17.4).
Proposition 28.10 For k 2 N the representation 2k of SU.2/ factors through a representation
N 2k of SO.3/ ' SU.2/=fI g: The representations N 2k are mutually inequivalent and exhaust
\
SO.3/:
Proof: One readily verifies that 2k .x/ D I for x 2 fI g: Hence 2k factors through a
representation N 2k of SO.3/: Every invariant subspace of the representation space V2k of 2k is
2k .SU.2// invariant if and only if it is N 2k .SO.3// invariant. A non-trivial SO.3/-equivariant
map V2k ! V2l would also be SU.2/-equivariant. Hence the N 2k are mutually inequivalent.
\ assume that .; V / is an irreducible
Finally, to see that the representations N 2k exhaust SO.3/;
representation of SO.3/: Then ' WD ' is an irreducible representation of SU.2/; hence
equivalent to some n ; n 2 N: From ' D I on ker ' it follows that n D I on fI g; hence
n is even.
Lemma 29.1 Assume that G is connected, and let V; V 0 be two finite dimensional G-modules.
(a) Let W be a linear subspace of V: Then W is G-invariant if and only if W is g-invariant.
(b) The G-module V is irreducible if and only V is irreducible as a g-module.
(c) Let T W V ! V 0 be a linear map. Then T is G-equivariant if and only if T is g-equivariant.
99
(d) V and V 0 are isomorphic as G-modules if and only if they are isomorphic as g-modules.
From this it follows that 0 .x/ T D T .x/ for all x 2 exp g; and hence for x 2 Ge D G:
The reverse implication follows by a straightforward differentiation argument as in part (a) of
this proof.
(d): This follows immediately from (c).
Lemma 29.2 Let G be a connected compact Lie group, and let be a representation of G in a
finite dimensional Hilbert space V: Then is unitary if and only if
for all X 2 g:
Proof: We recall that W G ! GL.V / is a Lie group homomorphism. Hence for all X 2 g; t 2
R we have:
.exp tX/ D e t .X/ :
If is unitary, then .exp tX/ D .exp. tX//; hence
e t .X/ D e t .X/
: (39)
Differentiating this relation at t D 0 we find (38). Conversely, if (38) holds, then (39) holds for
all X; t and it follows that .x/ is unitary for x 2 exp g: This implies that .x/ is unitary for
x 2 Ge D G:
It will turn out to be convenient to extend representations of g to its complexification gC :
If E is a real linear space, its complexification EC is defined as the real linear space E R C;
equipped with the complex scalar multiplication .v z/ D v z: We embed E as a real
linear subspace of EC by the map v 7! v 1: Then EC D E iE as a real linear space. In terms
of this decomposition, the complex scalar multiplication is given in the obvious fashion. If g is a
real Lie algebra, then its complexification gC is equipped with the complex bilinear extension of
the Lie bracket. Thus, gC is a complex Lie algebra.
100
Any representation of g in a complex vector space V has a unique extension to a (complex)
representation of gC in V I this extension, denoted C ; is given by
C .X C iY / D .X/ C i.Y /;
for X; Y 2 g:
Lemma 29.3 Let V; V 0 be g-modules, and let W V a (complex) linear subspace, and T W
V ! V 0 a (complex) linear map.
(a) The space W is g-invariant if and only if it is gC -invariant.
(b) V is irreducible as a g-module if and only if it is so as a gC -module.
(c) T is g-equivariant if and only if it is gC -equivariant.
(d) V and V 0 are isomorphic as g-modules if and only if they are isomorphic as gC -modules.
Proof: Left to the reader.
Example 29.4 The Lie algebra su.2/ of SU.2/ consists of complex 22 matrices A 2 M.2; C/;
satisfying trA D 0 and A D A: It follows from this that i su.2/ is the real linear subspace
of M.2; C/ consisting of matrices A with trA D 0 and A D A: In particular, we see that
su.2/ \ i su.2/ D f0g: Therefore, the embedding su.2/ ,! M.2; C/ extends to a complex linear
embedding
j W su.2/C ,! M.2; C/:
Clearly, the image of j is contained in the Lie algebra of SL.2; C/; which is given by
sl.2; C/ D fA 2 M.2; C/ j trA D 0g:
On the other hand, if A 2 sl.2; C/; then 21 .A A / belongs to su.2/ and 21 .A C A / belongs
to i su.2/I summing these elements, we see that A 2 j.su.2/C /: Therefore, j is an isomorphism
from su.2/C onto sl.2; C/; via which we shall identify from now on.
30 Representations of sl(2,C)
It follows from the discussion in the previous section that the SU.2/-module Pn .C2 /; for n 2
N; carries a natural structure of sl.2; C/-module. The associated representation of sl.2; C/ in
Pn .C2 / equals .n /C ; the complexification of n : We shall now compute this structure in
terms of the basis p0 ; : : : ; pn of Pn .C2 / given by
pj .z/ D z1j z2n j
; .z 2 C2 /:
Let p 2 Pn .C2 /: Then we recall that, for x 2 SU.2/; n .x/p.z/ D p.x 1
z/; z 2 C2 : It
follows from this that, for 2 su.2/;
d
t
n ./p.z/ D p.e z/ ;
dt t D0
101
hence, by the chain rule
@p @p
n ./p.z/ D .z/. z/1 C .z/. z/2 :
@z1 @z2
The expression on the right-hand side is complex linear in I hence it also gives p D .n /C ./p
for 2 sl.2; C/: Thus, we obtain, for 2 sl.2; C/ and p 2 Pn .C2 /;
@ @
p D .z/1 C .z/2 p: (40)
@z1 @z2
We shall now compute the action of the basis H; X; Y of sl.2; C/ given by
1 0 0 1 0 0
H D ; XD ; Y D :
0 1 0 0 1 0
By a straightforward computation we see that
H; X D 2X; H; Y D 2Y; X; Y D H: (41)
Definition 30.1 Let l be a Lie algebra. By a standard sl.2/-triple in l we mean a collection of
linear independent elements H; X; Y 2 l satisfying the relations (41).
Remark 30.2 Let l be a complex Lie algebra. Then the complex linear span of an sl.2/-triple
in l is a Lie subalgebra isomorphic to sl.2; C/:
Substituting H; X and Y for in (40), we obtain, for p 2 Pn .C2 /;
@ @ @ @
H p D z1 C z2 p; XpD z2 p; YpD z1 p: (42)
@z1 @z2 @z1 @z2
By a straightforward computation we now see that the action of the triple H; X; Y on the basis
element pj is given by
Hpj D .n 2j /pj ; Xpj D jpj 1; Ypj D .j n/pj C1 :
For the matrices of the action of H; X; Y on Pn.C2 / relative to the basis p0 ; : : : ; pn we thus find
0 1
n 0 ::: 0
B :: C
B 0 n 2 : C
mat.H / D B : C;
@ :: : : : ::: A
0 ::: ::: n
and
0 1 0 1
0 10 ::: 0 0 ::: ::: 0
B 0 0 2 ::: 0 C B :: C
B C B n 0 : C
B :: :: :: :: C B C
mat.X/ D B : : : : C mat.Y / D B 0 1 n 0 C:
B C B :: : : : : :: C
@ 0 n A @ : : : : A
0 ::: ::: 0 0 ::: 0 1 0
These matrices will guide us through the proof of the following theorem.
102
Theorem 30.3 Every irreducible finite dimensional sl.2; C/-module is isomorphic to Pn .C2 /;
for some n 2 N:
Remark 30.4 From the above theorem we deduce again, using Lemmas 29.1 and 29.3, that
every irreducible continuous finite dimensional representation of SU.2/ is equivalent to n ; for
some n 2 N:
The proof of the the above theorem will be given in the rest of this section. Let V be an
irreducible finite dimensional sl.2; C/-module.
Given 2 C; we shall write V WD ker.H I /: This space is non-trivial if and only if is
an eigenvalue for the action of H on V:
XV VC2 ; Y V V 2 :
Lemma 30.7
(a) The vectors vk D Y k v; 0 k n; form a basis for V:
(b) The eigenvalue equals n D dimV 1:
(c) For every 0 k n;
103
(d) The primitive vectors in V are the non-zero multiples of v0 :
Proof: We first prove (c) for all k 2 N (but note that vk D 0 for k > n). It follows from repeated
application of Lemma 30.5 that vk 2 V 2k ; hence H vk D . 2k/vk : We prove the second
assertion of (c) by induction. Since v0 D v is primitive, the second assertion of (c) holds for
k D 0: Let now k > 0 and assume that the assertion has been established for strictly smaller
values of k: Then
Xvk D XY vk 1
D YXvk 1 C X; Y vk 1
D YXvk 1 C H vk 1
D .k 1/. .k 2//Y vk 2 C . 2.k 1//vk 1
D k. k C 1/vk 1
Corollary 30.8 Let V and V 0 be two irreducible finite dimensional sl.2; C/-modules. Then
V ' V 0 if and only if dimV D dimV 0 : Moreover, if v and v 0 are primitive vectors of V and V 0 ;
respectively, then there is a unique isomorphism T W V ! V 0 mapping v onto v 0 :
Proof: Clearly if V ' V 0 then V and V 0 have equal dimension. Conversely, assume that
dimV D dimV 0 D n and that v and v 0 are primitive vectors of V and V 0 respectively. Then by
the above lemma, the vectors vk D Y k v; 0 k n form a basis of V: Similarly the vectors
vk0 D Y k v 0 ; 0 k n form a basis of V 0 : Any intertwining operator T W V ! V 0 that maps v
onto v 0 must map the basis vk onto the basis vk0 ; hence is uniquely determined. Let T W V ! V 0
be the linear map determined by T vk D vk0 ; for 0 k n: Then T is a linear bijection.
Moreover, by the above lemma we see that T intertwines the actions of H; X; Y on V and V 0 : It
follows that T is equivariant, hence V ' V 0 :
Completion of the proof of Theorem 30.3: The space Pn .C2 / is an irreducible sl.2; C/-
module, of dimension n C 1: Hence if V is an irreducible sl.2; C/-module of dimension m 1;
then V ' Pn .C2 /; with n D m 1:
104
31 Roots and weights
Let t be a finite dimensional commutative real Lie algebra, and let .; V / be a finite dimensional
representation of t in V:
Let tC denote the space of complex linear functionals on tC : Note that t ; the space of real
linear functionals on t may be identified with the space of 2 tC that are real valued on t: Thus,
t is viewed as a real linear subspace of tC : Accordingly i t equals the space of 2 tC such that
jt has values in i R:
If 2 tC ; then we define the following subspace of V W
\
V D ker..H / .H /I /: (43)
H 2t
In other words, V consists of the space of v 2 V such that .H /v D .H /v for all H 2 t: If
V 0; then is called a weight of t in V; and V is called the associated weight space. The set
of weights of t in V is denoted by ./:
Proof: Let 2 ./: The endomorphism T commutes with .H / hence leaves the eigenspace
ker..H / .H // invariant, for every H 2 t: Hence T leaves the intersection V of all these
spaces invariant.
Lemma 31.2 The set ./ is a non-empty finite subset of tC : Assume that .X/ is diagonaliz-
able for every X 2 t: Then M
V D V : (44)
2./
Proof: Fix a basis X1 ; : : : ; Xn of t: The endomorphism .X1 / has at least one eigenvalue, say
1 ; with corresponding eigenspace E1 V: Since t is commutative, this eigenspace is invariant
under the action of t: Proceeding by induction on dimt; we obtain a sequence of non-trivial
subspaces En En 1 E1 such that Xj acts by a scalar j on Ej ; for each 1 j n:
Define 2 tC by .Xj / D j ; then En V ; hence 2 ./: This establishes the first
assertion.
If .X/ diagonalizes, for every X 2 t; then, in particular, V admits a decomposition of
eigenspaces for the endomorphism .X1 /: Each of these eigenspaces is invariant under t: There-
fore, by induction on dimt there exists a direct sum decomposition V D V1 VN such that
Xj acts by a scalar ij on Vi ; for all 1 i N and 1 j n: Let i 2 tC be defined by
i .Xj / D ij ; for 1 i N: Then ./ D f1 ; : : : ; N g: Moreover, one readily verifies that,
for 2 ./; V D j WDj Vj : Hence, (44) follows.
105
For the final assertion, we observe that by finite dimensionality of V the set ./ is finite.
Hence, there exists a X0 2 t such that . /.X0 / 0 for all ; 2 ./ with : For
2 ./; let P W V ! V be the projection along the remaining summands in (44). We claim
that Y
P D ..X0 / .X0 // 1 ..X0 / .X0 //:
2./nfg
Indeed this is readily checked on each of the summands V of the decomposition in (44), for
2 ./:
It follows from the above formula for P that P .W / W: Hence, P .W / W \ V ; and
the final assertion follows.
Assumption: In the rest of this section we assume that G is a compact Lie group, with Lie
algebra g:
Definition 31.3 A torus in g is by definition a commutative subalgebra of g: A torus t g is
called maximal if there exists no torus of g that properly contains t:
From now on we assume that t is a fixed maximal torus in g:
From Lemma 31.2 we see that . / is a non-empty finite subset of tC :
Let .; V / be a finite dimensional continuous representation of G: Then the map W G !
GL.V / is a homomorphism of Lie groups. Let D Te : Then W g ! End.V / is a Lie
algebra homomorphism, or, differently said, a representation of g in V: The homomorphism
has a unique extension to a complex Lie algebra homomorphism from gC into End.V / (we
recall that V is a complex linear space by assumption). This extension is called the induced
infinitesimal representation of gC in V:
106
If V is equipped with a G-invariant inner product, then for all ; 2 . / with we
have V ? V :
Proof: There exists a G-invariant inner product on V I assume such an inner product h ; i to be
fixed. Then maps G into U.V /; the associated group of unitary transformations. It follows
that maps g into the Lie algebra u.V / of U.V /; which is the subalgebra of anti-Hermitian
endomorphisms in End.V /: It follows that for X 2 g the endomorphism .X/ is anti-Hermitian,
hence diagonalizable with imaginary eigenvalues. The proof is now completed by application of
Lemma 31.2.
If A 2 End.g/; then we denote by AC the complex linear extension of A to gC : Obviously the
map A 7! AC induces a real linear embedding of End.g/ into End.gC / WD EndC .gC /: Accord-
ingly we shall view End.g/ as a real linear subspace of the complex linear space End.gC / from
now on. Thus, we may view Ad as a representation of G in the complexification gC of g: The
associated infinitesimal representation is the adjoint representation ad of gC in gC : The associ-
ated collection . ad/ of weights contains the weight 0: Indeed the associated weight space gC0
equals the centralizer of t in gC ; which in turn equals tC ; by Lemma 31.4. Hence:
gC0 D tC :
Definition 31.6 The weights of ad in gC different from 0 are called the roots of t in gC I the set
of these is denoted by R D R.gC ; t/: Given 2 R; the associated weight space gC is called a
root space.
From Lemma 31.5 we now obtain the so called root space decomposition of gC ; relative to the
torus t:
Corollary 31.7 The collection R D R.gC ; t/ of roots is a finite subset of i t : Moreover, we have
the following direct sum of vector spaces:
M
gC D tC gC : (45)
2R
Example 31.8 The Lie algebra g D su.2/ has complexification sl.2; C/; consisting of all com-
plex 2 2 matrices with trace zero. Let H; X; Y be the standard basis of sl.2; C/I i.e.
1 0 0 1 0 0
H D ; XD ; Y D :
0 1 0 0 1 0
107
We recall that, by definition, the center z D zg of g is the ideal ker adI i.e., it is the space of
X 2 g that commute with all Y 2 g:
Lemma 31.9 The center of g is contained in t and equals the intersection of the root hyper-
planes: \
zg D ker :
2R
Proof: The center of g centralizes t in particular, hence is contained in t; by Lemma 31.4. Let
H 2 t and assume that H centralizes gI then H centralizes gC ; hence every root space of gC : This
implies that .H / D 0 for all 2 R: Conversely, if H 2 t is in the intersection of all the root
hyperplanes, then H centralizes tC and every root space gC : By the root space decomposition it
then follows that H 2 z: This establishes the characterization of the center.
If z D 0; then the root hyperplanes ker . 2 R/ have a zero intersection in t: This implies
that the set R i t spans the real linear space i t :
Lemma 31.10 Let .; V / be a finite dimensional representation of gC : Then for all 2 ./
and all 2 R [ f0g we have:
.gC /V VC :
In particular, if C ./; then .gC / anihilates V :
Hence .X/v 2 VC : If C is not a weight of ; then VC D 0 and it follows that
.X/v D 0:
gC ; gC gC.C / :
Proof: This follows from the previous lemma applied to the adjoint representation.
P We shall write ZR for the Z-linear span of R; i.e., the Z-module of elements of the form
2R n ; with n 2 Z:
In the following corollary we do not assume that comes from a representation of G:
108
Corollary 31.12 Let .; V / be a finite dimensional representation of gC : Then
M
W WD V (46)
2./
Proof: By Lemma 31.2 the set ./ is non-empty and finite, and therefore W is a non-trivial
subspace of V: From Lemma 31.10 we see that W is gC -invariant. If is irreducible, then
W D V: To establish the last assertion we define an equivalence relation on ./ by
() 2 ZR: If S is a class for ; then VS D 2S V is a non-trivial gC -invariant
subspace of V; by Lemma 31.10. Hence VS D V and it follows that S D ./:
Remark 31.13 If g has trivial center, then the above result actually holds for every finite dimen-
sional V -module. To see that a condition like this is necessary, consider g D R; the Lie algebra
of the circle. Define a representation of g in V D C2 by
0 x
.x/ D :
0 0
Lemma 31.14 Let t be a maximal torus in g; and R the associated collection of roots. If 2 R
then 2 R:
Proof: Let be the conjugation of gC with respect to the real form g: That is: .X C iY / D
X iY for all X; Y 2 g: One readily checks that is an automorphism of gC ; considered as a
real Lie algebra (by forgetting the complex linear structure). Let 2 R; and let X 2 gC : Then
for every H 2 t;
For the latter equation we used that has imaginary values on t: It follows that 2 R and that
maps gC into gC (in fact is a bijection between these root spaces; why?).
We recall that we identify i t with the real linear subspace of tC consisting of such that jt
has values in i RI the latter condition is equivalent to saying that jit is real valued. One readily
verifies that the restriction map 7! jit defines a real linear isomorphism from i t onto the
real linear dual .i t/ : In the following we shall use this isomorphism to identify i t with .i t/ :
Now R is a finite subset of .i t/ n f0g: Hence the complement of the hyperplanes ker i t;
for 2 R is a finite union of connected components, which are all convex. These components
are called the Weyl chambers associated with R: Let C be a fixed chamber. By definition every
109
root is either positive or negative on C: We define the system of positive roots RC WD RC .C/
associated with C by
RC D f 2 R j > 0 on Cg:
By what we said above, for every 2 R; we have that either or belongs to RC ; but not
both. It follows that
R D RC [ . RC / (disjoint union). (47)
We writePNRC for the subset of ZR consisting of the elements that can be written as a sum
of the form 2RC n ; with n 2 N:
Lemma 31.15 NRC \ . NRC / D 0:
Proof: Let 2 NRC : Then 0 on C; the chamber corresponding to RC : If also 2 NRC ;
then 0 on C as well. Hence D 0 on C: Since C is a non-empty open subset of i t ; this
implies that D 0:
Lemma 31.18 Let V be a finite dimensional gC -module. Then V has a highest weight vector.
Proof: We define the gC -submodule W of V as the sum of the tC -weight spaces, see Corollary
31.12.
Let C be the positive chamber determining RC : Fix X 2 C: Then .X/ > 0 for all 2 RC :
We may select 0 2 ./ such that the real part of .X/ is maximal. Then 0 C ./ for
all 2 RC : By Lemma 31.10 this implies that .gC /V V0 C D 0 for all 2 RC : Hence
gC
C annihilates V0 : Thus, every non-zero vector of V0 is a highest weight vector.
110
Definition 31.19 Let V be a finite dimensional gC -module. A vector v 2 V is called cyclic if it
generates the gC -module V; i.e., V is the smallest gC -submodule containing v:
Proposition 31.20 Let V be a finite dimensional gC -module and let v 2 V be a cyclic highest
weight vector.
(a) There exists a (unique) 2 .V / such that v 2 V : Moreover, V D Cv:
(b) The space V is equal to the span of the vectors v and .X1 / .Xn /v; with n 2 N and
Xj 2 gC ; for 1 j n:
(c) Every weight 2 .V / is of the form ; with 2 NRC :
(d) The module V has a unique maximal proper submodule W:
(e) The module V has a unique non-trivial irreducible quotient.
Proof: The first assertion of (a) follows from the definition of highest weight vector. We define
an increasing sequence of linear subspaces of V inductively by V0 D Cv and VnC1 D Vn C
.gC /Vn : Let W be the union of the spaces Vn : We claim that W is an invariant subspace of
V: To establish the claim, we note that by definition we have .gC /Vn VnC1 I hence W is gC
invariant. The space V0 is t- and gC
C -invariant; by induction we will show that the same holds for
C
Vn : Assume that Vn is t- and gC -invariant, and let v 2 Vn ; Y 2 gC : Then for H in t we have
H Y v D Y H v C H; Y v: Now v 2 Vn and by the inductive hypothesis it follows that H v 2 Vn :
Hence Y H v 2 VnC1 : Also H; Y 2 gC and it follows that H; Y v 2 VnC1 : We conclude that
H Y v 2 VnC1 : It follows from this that
Hence VnC1 is gC C -invariant. This establishes the claim that W is a gC -invariant subspace of V:
Since W contains the cyclic vector v; it follows that W D V: Hence, (b) follows. Let
w D .Y1 / .Yn /v; with n 2 N; Yj 2 gC. j / ; j 2 RC : Then w belongs to the weight
space V ; where D 1 C C n 2 NRC : Since v and such elements w span W D V;
we conclude that every weight in .V / is of the form with 2 NRC : This establishes
(c). Moreover, it follows from the above description that V equals the vector sum of Cv and
V ; where V denotes the sum of the weight spaces V with 2 .V / n fg: This implies that
V D Cv; whence the second assertion of (a).
111
We now turn to assertion (d). Let U be a submodule of V: In particular, U is a tC -invariant
subspace. Let .U / be the collection of 2 .V / for which U WD U \ V 0: In view of
Lemma 31.2, U is the direct sum of the spaces U ; for 2 .U /: If U is a proper submodule,
then U D 0; hence .U / .V / n fg: It follows that the vector sum W of all proper
submodules satisfies .W / .V / n fg hence is still proper. Therefore, V has W as unique
maximal submodule.
The final assertion (e) is equivalent to (d). To see this, let p W V ! V 0 be a surjective gC -
module homomorphism onto a non-trivial gC -module. Then U 7! p 1 .U / defines a bijection
from the collection of proper submodules of V 0 onto the collection of proper submodules of
V containing ker p: It follows that V 0 is irreducible if and only if ker p is a proper maximal
submodule of V: The equivalence of (d) and (e) now readily follows.
Corollary 31.21 Let V be a finite dimensional irreducible gC -module. Then V has a highest
weight vector v, which is unique up to a scalar factor. Let be the weight of v: Then assertions
(a) - (c) of Proposition 31.20 are valid.
Proof: It follows from Lemma 31.18 that V has a highest weight vector. Let v be any highest
weight vector in V and let be its weight. By irreducibility of V; the vector v is cyclic. Hence
all assertions of Proposition 31.20 are valid.
Let w be a second highest weight vector and let be its weight. Then all assertions of
Proposition 31.20 are valid. Hence 2 NRC and 2 NRC ; from which 2
NRC \ . NRC / D f0g: It follows that D I hence w 2 V D Cv:
Remark 31.22 For obvious reasons the above weight is called the highest weight of the
irreducible gC -module V; relative to the choice RC of positive roots.
The following theorem is the first step towards the classification of all finite dimensional
irreducible representations of gC :
Theorem 31.23 Let V and V 0 be irreducible gC -modules. If V and V 0 have the same highest
weight (relative to RC ), then V and V 0 are isomorphic (i.e., the associated gC -representations
are equivalent).
Proof: We denote the highest weight by and fix associated highest weight vectors v 2 V n f0g
and v 0 2 V0 n f0g: We consider the direct sum gC -module V V 0 and denote by W the smallest
gC -submodule containing the vector w WD .v; v 0/: Then w is a cyclic weight vector of W; of
weight :
Let p W V V 0 ! V be the projection onto the first component, and p 0 W V V 0 ! V 0 the
projection onto the second. Then p and p 0 are gC -module homomorphisms. Since p.w/ D v; it
follows that pjW is surjective onto V: Similarly, p 0 jW is surjective onto V 0 : It follows that V; V 0
are both irreducible quotients of W; hence isomorphic by Proposition 31.20 (e).
Remark 31.24 In the above proof it is easy to deduce that in fact W is irreducible, and pjW and
p 0 jW are isomorphisms from W onto V and V 0 ; respectively.
112
32 Conjugacy of maximal tori
We retain the notation of the previous section. In this section we shall investigate to what extent
the collection R D R.gC / depends on the choice of the maximal torus t: An element X 2 tC
will be called regular if .X/ 0 for all 2 R: The set of regular elements in t and tC will be
reg
denoted by treg and tC ; respectively. Since R is finite, treg is an open dense subset of tI similarly
reg
tC is an open dense subset of tC :
Lemma 32.1 Let t be a maximal torus in g; and let X 2 t: Then the following statements are
equivalent.
(a) X 2 treg I
(b) ker. ad.X// D tI
(c) with respect to any G-invariant inner product on g we have t D im. ad.X//? I
(d) with respect to some G-invariant inner product on g we have t D im. ad.X//? I
Since X is regular, .X/ 0 for all ; and it follows that Y D 0 for all 2 R: Hence
Y 2 g \ tC D t: This implies ker. ad.X// tI the converse inclusion is obvious, hence (b)
follows.
Next, we assume that (b) holds. Since ad.X/ is anti-symmetric with respect to any invariant
inner product, it follows that im. ad.X//? D ker. ad.X//: The latter space equals t by (b). Hence
(c) follows.
That (c) implies (d) is obvious. Now assume that (d) holds. Then it follows that ad.X/
induces a linear automorphism of g=t: All eigenvalues of a linear automorphism must be different
from zero, hence .X/ 0 for all 2 R:
If g 2 G; then Ad.g/ is an automorphism of the Lie algebra gI hence Ad.g/t is a maximal
torus in g: The following result asserts that all maximal tori of g arise in this way.
Lemma 32.2 Let t; t0 be two maximal tori in g: Then there exists a g 2 G such that
t0 D Ad.g/t:
Proof: By the method of averaging over G we see that there exists a G-invariant positive definite
inner product on gI select such an inner product h ; i: Moreover, select regular elements X 2 t
and Y 2 t0 : Then by Lemma 32.1 we see that t equals the centralizer of X in g: We consider the
smooth function f W G ! R given by
f .x/ D hAd.x/X ; Y i:
113
By compactness of G; the continuous function f attains a minimal value at a point x0 2 G: It
follows that for every Z 2 g the function t 7! f .x0 exp t Z/ has a minimum at t D 0; hence
d
0D f .x0 exp t Z/ D hAd.x0 /Z; X ; Y i D h ad.X/.Z/ ; Ad.x0 / 1 Y i:
dt t D0
By Lemma 32.1 we see that ad.X/ maps g onto t? : Hence Ad.x0 / 1 Y 2 .t? /? D t: It follows
from this that the maximal torus t00 D Ad.x0 /t contains Y I obviously t00 is contained in the
centralizer of Y; which equals t0 ; by Lemma 32.1. By maximality of t00 it follows that t0 D t00 D
Ad.x0 /t:
If g 2 G; then Ad.g/ is an automorphism of the Lie algebra g: More generally we now
consider an automorphism ' of the Lie algebra gI its complex linear extension, also denoted by
' is an automorphism of the complex Lie algebra gC : If t is a maximal torus, then t0 D '.t/ is a
maximal torus as well. The map tC ! t0C given by 7! ' 1 is a linear isomorphism, which
we again denote by ': With this notation we have:
Lemma 32.3 Let ' be an automorphism of the Lie algebra g: If t is a maximal torus in g; then
t0 D '.t/ is a maximal torus in g as well. Moreover, the induced linear isomorphism ' W tC ! t0C
maps R D R.gC ; t/ bijectively onto R0 D R.gC ; t0 /: Finally, if 2 R; then
'.gC / D gC'./ :
H 0 ; '.X/ D '.' 1
.H 0 /; X/ D '..' 1
.H 0 /X/ D './.H 0 /'.X/:
From this we see that './ 2 R0 and '.gC / gC'./ : The proof is completed by applying the
same reasoning to the inverse of ':
Corollary 32.4 Let R; R0 be the collections of roots associated with two maximal tori t; t0 of g:
Then there exists a bijective linear map from i t onto i t0 which maps R onto R0 :
Proof: By Lemma 32.2 there exists a g 2 G such that Ad.g/t D t0 : The map ' D Ad.g/ is
an automorphism of g: By Lemma 32.3 the induced isomorphism from tC onto t0C satisfies all
requirements.
114
subgroup of GL.g/: Its Lie algebra is a sub Lie algebra of End.g/; equipped with the commutator
bracket.
A derivation of g is by definition a linear map D 2 End.g/ such that
One readily sees that the space Der.g/ of all derivations of g is a Lie subalgebra of End.g/:
Proof: Let D be an element in the Lie algebra of Aut.g/: Then exp.tD/ 2 Aut.g/ for all t 2 R:
Let X; Y 2 g; then it follows that
e tD X; Y D e tD X; e tD Y :
Hence c is constantly equal to c.0/ D X; Y : It follows from this that e tD 2 Aut.g/ for all
t 2 R; hence D belongs to the Lie algebra of Aut.g/:
Corollary 33.2 The homomorphism ad maps g into Der.g/: If X 2 g; then e adX is an automor-
phism of the Lie algebra g:
Proof: The first assertion follows from the Jacobi identity. The last statement is now a conse-
quence of the above lemma; indeed e D 2 Aut.g/; for D 2 Der.g/:
The subgroup of Aut.g/ generated by e ad.X/ ; X 2 g is called the group of interior automor-
phisms of g; notation: Int.g/: Its Lie algebra equals ad.g/; see Section 7.
115
Lemma 34.1 The Killing form is symmetric. Morever, if ' 2 Aut.g/; then
Finally,
B.Z; X; Y / D B.X; Z; Y / .X; Y; Z 2 g/: (49)
Proof: If A; B are endomorphisms of a linear space, it is well known that AB BA has trace 0:
Hence tr. ad.X/ ad.Y // D tr. ad.Y / ad.X//; for X; Y 2 g; and the symmetry of B follows.
If ' is a Lie algebra automorphism of g; then it follows that ' adX D ad.'.X// ': Hence
ad.'.X// D ' ad.X/ ' 1 : Using this and conjugation invariance of the trace (48) follows.
Let t 2 R; then e t adZ 2 Aut.g/I thus (48) holds with e t adZ inserted for ': Differentiation of
the resulting identity with respect to t at t D 0 yields (49).
The latter identity can also be derived algebraically, as follows. We note that ad.Z; X/ D
ad.Z/ ad.X/ ad.X/ ad.Z/; hence
The identity (49) is known as invariance of the Killing form. If v is a linear subspace of g;
then by v? we shall denote its orthocomplement with respect to B; i.e., the collection of Y 2 g
such that B.X; Y / D 0 for all X 2 v: Note that from the invariance of B the following lemma is
an immediate consequence.
Lemma 35.1 Let g be compact. Then the Killing form B is negative semi-definite. Moreover,
g? D z:
116
Proof: We may assume that g is the Lie algebra of the compact group G: The representation
Ad of G in gC is unitarizable, hence there exists a positive definite inner product on gC that is
Ad.G/-invariant. With respect to this inner product we have Ad.G/ U.gC /:
Since ad is the infinitesimal representation of g in gC associated with Ad; it follows that
ad.g/ u.gC /: Hence, ad.X/ is an anti-symmetric Hermitian endomorphism of gC ; for X 2 g:
This implies that adX has a an orthonormal basis of eigenvectors and imaginary eigenvalues.
Hence ad.X/2 has the same orthonormal basis of eigenvectors with eigenvalues 0: It follows
that B.X; X/ D tr ad.X/2 0: Hence, B is negative semi-definite. Moreover, if B.X; X/ D 0
then tr ad.X/2 D 0 and it follows that all eigenvalues of ad.X/2 ; hence of ad.X/ are zero.
Hence, ad.X/ D 0: This shows that g? z: The converse inclusion is obvious.
If v; w are subspaces of g; then by v; w we denote the subspace of g spanned by the elements
X; Y ; where X 2 v; Y 2 w: If v; w are ideals, then v; w is an ideal of g: Indeed, this follows
by a straightforward application of the Jacobi identity. In particular
Dg WD g; g
Lemma 35.3 Let g be a compact Lie algebra. Then every ideal of g has a complementary ideal.
Proof: As in the proof of Lemma 35.1 there exists a positive definite inner product h ; i on
g with respect to which Ad.G/ O.g/: It follows that ad.g/ o.g/; or, equivalently, that
hX; U ; V i D hU ; X; V i; for all X; U; V 2 g: By this property, if a g is an ideal, then
a? is an ideal; moreover, g D a a? :
Lemma 35.4 Let g have the property that every ideal has a complementary ideal. Then
g D z Dg:
Proof: The ideal Dg has a complementary ideal, say a: Since obviously g; a Dg; we have
g; a a \ Dg D 0; from which we conclude that a z: It follows that g D z C Dg:
The ideal z has a complementary ideal, say b: Thus, g D z b: Now Dg D g; g
g; z C g; b b; from which we conclude that z \ Dg D 0:
117
Theorem 35.5 The following assertions are equivalent.
(a) g is compact
(b) g D z Dg and B is negative definite on Dg:
(c) There exists a subalgebra g0 g such that g D z C g0 and such that B is negative definite
on g0 :
Finally, if g0 is as in (c) then g0 D Dg:
Proof: First, assume that (a) is valid. Then g D z Dg by Lemma 35.4. By Lemma 35.1 it
follows that B < 0 on Dg: Hence (b). The implication (b) ) (c) is obvious. The implication (c)
) (a) and the final assertion will be established in the following lemma.
Lemma 35.6 Let the Killing form of g be negative definite. Then Aut.g/ is compact. Moreover,
ad is a Lie algebra isomorphism from g onto Der.g/: In particular it follows that g is compact
and has trivial center.
Proof: Let O.g/ denote the group of invertible transformations of g that are orthogonal relative
to the positive definite inner product B: Then O.g/ is compact. From (48) it follows that the
closed subgroup Aut.g/ of GL.g/ is contained in the compact group O.g/; hence is compact.
By definition of the Killing form, ker ad g? I since B is non-degenerate, it follows that ad
is an injective Lie algebra homomorphism. It follows from the Jacobi identity that ad maps g
into Der.g/:
If D 2 Der.g/; then for X 2 g we have that D ad.X/.Y / D ad.DX/Y C ad.X/ D.Y /
for Y 2 g: Hence
D; ad.X/ D ad.DX/: (51)
It follows that ad.g/ is an ideal in Der.g/:
Now Der.g/ is the Lie algebra of the compact group Aut.g/; see Proposition 33.1. It follows
that Der.g/ is compact. By application of Lemma 35.3 it follows that ad.g/ has a complementary
ideal b in Der.g/: Let D 2 b: Then D commutes with ad.g/; hence from (51) we see that
ad.DX/ D 0; whence DX D 0 for all X 2 g: Hence D D 0: We conclude that b D 0; hence
ad.g/ D Der.g/:
It follows that ad is an isomorphism from g onto Der.g/I the latter is the Lie algebra of the
compact group Aut.g/: Hence g is compact.
Completion of the proof of Theorem 35.5: Assume that (c) holds. Then g0 has negative
definite Killing form, hence is compact. Since z D ker ad g? it follows that z \ g0 D 0:
Hence, g D z g0 as linear spaces. Since obviously z; g0 D 0; the mentioned direct sum is a
direct sum of Lie algebras.
Let G 0 be a compact Lie group with algebra isomorphic to g0 : Let n D dimz: Then z ' Rn
as abelian Lie algebras. Hence, the compact torus T WD Rn =Zn has Lie algebra isomorphic z:
It follows that the compact group G WD T G 0 has Lie algebra isomorphic to z g0 D g: (a)
follows.
Finally, let g0 be as in (c). Then by the above reasoning, g D z g0 as Lie algebras. It follows
that Dg g0 ; g0 g0 : Now apply (b) to conclude that Dg D g0 :
118
The Lie algebra g is called simple if it is not abelian and has no ideals besides 0 and g: It is
called semisimple if it is a direct sum of simple ideals.
Lemma 35.7 Let g be semisimple, then z D 0 and Dg D g:
Proof: Let g D g1 gn be a decomposition into simple ideals. We observe that each ideal gj
is non-abelian, hence gj ; gj is a non-trivial ideal in gj : Since the latter is simple, we conclude
that Dgj D gj : Since the gj mutually commute, it follows that Dg D Dg1 C C Dgn D g: If
X belongs to the center of g; write X D X1 C C Xn ; according to the decomposition (52).
Then X commutes with gi and each Xj ; for j i; commutes with gi : Hence, Xi commutes
with gi as well. Hence X belongs to the center zi of gi : This center is an ideal different from gi ;
since gi is not abelian. Since gi is simple, zi D 0: We conclude that X D 0: Hence, z D 0:
Lemma 35.9 Let g be a finite dimensional Lie algebra, and let a be a simple ideal of g: If
g D g1 : : : gn is a direct sum of ideals, then there exists a unique 1 j n such that
gj a:
Proof: We note that g; a a since a is an ideal, and g; a a; a D a since a is simple.
Hence g; a D a: From the direct sum decomposition it now follows that
a D g1 ; a C C gn ; a:
Hence there exists a j such that gj ; a 0: Since a is simple and gj ; a is an ideal in a we must
have gj ; a D a: This implies that a D a; gj gj : Of course j is uniquely determined by the
latter property.
119
Lemma 35.10 Let g be semisimple, and let S be the collection of simple ideals in g: Every ideal
a g is the direct sum of the ideals from S that are contained in a: In particular, g is the direct
sum of the ideals from S:
Proof: We may express g as a direct sum of simple ideals of the form (52). If a 2 S then, by
the previous lemma, a gj for some j: Since gj is simple, it follows that a D gj : We conclude
that S D fg1 ; : : : ; gn g:
Let now b g be any ideal. We will show that b is the direct sum of the simple ideals from
S.b/ WD fa 2 S j a bg by induction on #S: First, assume that #S D 1: Then g is simple,
hence b D 0 or b D g and the result follows. Now assume that #S > 1 and that the result has
been established for g with S of strictly smaller cardinality. If b D 0 there is nothing to prove. If
b 0; then g; b 0 since z D 0: It follows that gj ; b 0 for some j: But gj ; b is an ideal
in the simple algebra gj ; hence gj D gj ; b b: Let g0 be the direct sum of the ideals from the
non-empty set S.b/; and let g00 be the direct sum of the ideals from S n S.b/: Then g0 b and
g D g0 g00 ; hence b D g0 .b \ g00 /: Now b \ g00 is an ideal in the semisimple algebra g00 : By
the induction hypothesis, b \ g00 is the direct sum of the ideals from S contained in both b and
g00 : This set is empty, hence b \ g00 D 0 and the result follows.
(c) gC ; gC CH :
120
Proof: Let X 2 gC and Y 2 gC : Then by invariance of the Killing form we have, for all
H 2 tC ;
.H / C .H / B.X; Y / D B.H; X; Y / C B.X; H; Y / D 0:
From this, (a) follows.
Let now ; X; Y be as in (b). Then X; Y 2 gC0 D tC ; by Corollary 31.11 and Lemma 31.4.
Moreover, for all H 2 tC we have
Hence, (b).
Finally, for (c) we note that (b) implies that X; Y ? ker ; relative to B; for X 2 gC and
Y 2 gC : In view of (53) this implies that gC ; gC CH :
Let W gC ! gC be the conjugation with respect to the real form g of gC : Thus, D I on
g and D I on i g: We recall that .gC / D gC ; for all 2 R; see proof of Lemma 31.14.
We denote the positive definite inner product B. ; /jg by h ; i and extend it to a Hermitian
positive definite inner product on gC : Then
hX ; Y i D B.X; Y /; .X; Y 2 gC /:
Lemma 36.3 Let 2 R: Then there exists a X 2 gC such that H ; X and Y WD X form
a standard sl.2; C/-triple.
B.X; Y / > 0:
121
Example 36.4 Let g D su.2/: Then gC D sl.2; C/ and the conjugation is given by A D
A ; where the star indicates that the Hermitian adjoint is taken. Let H; X; Y be the usual
standard triple in sl.2; C/: Thus, H is diagonal with entries C1; 1 and X is upper triangular
with 1 in the upper right corner, Y is lower triangular with 1 in the lower left corner. Then
t D i RH is a maximal torus in g: Moreover, R D f; g; where 2 i t is determined by
.H / D 2: Finally, X D Y; and we see that the above lemma with X D X gives us the
usual standard triple.
Proof: We observe that, for every X 2 l; the endomorphism ad.X/ of gC is anti-symmetric with
respect to h ; i: Indeed, this follows from invariance of the Killing form. Thus, if W V is a
ad.l/-invariant subspace, then so is W ? \ V: The lemma follows by repeated application of this
observation.
s WD gC CH gC
is isomorphic with sl.2; C/: Its intersection with g is isomorphic with su.2/: Finally, t WD i RH
is a maximal torus in s \ g and the associated root system is equal to fjt ; jt g:
and leave it to the reader to verify that V is invariant under the adjoint action of s: It follows
that V splits as a direct sum of irreducible s-modules. The decomposition of each irreducible
s-submodule in CH -weight spaces is compatible with the given weight space decomposition of
V: All weights of the irreducible representations of s belong to 12 Z0 ; with 0 D jCH : Thus,
if 2 R \ R; then jCH 2 21 0 ; from which we conclude that 2 R \ 12 Z: It follows that
R \ R D R \ 21 Z: Put
X
Vev D gC CH :
2R\Z
122
and let Vod d be the sum of the remaining root spaces in V: Then both Vev and Vod d are s -
invariant. The first of these spaces splits as a direct sum of irreducible s-modules all of whose
weights belong to Z0 : By the classification of irreducible sl.2; C/-modules it follows that each
of the irreducible summands has a zero weight space, which must be contained in CH : It follows
that Vev has only one irreducible summand, hence is irreducible. Since s Vev is an invariant
subspace, it follows that s D Vev : This implies that R \ Z D f; g and s D s :
It remains to be shown that Vod d is the zero space. Assume not. Then V has a CH -
weight of the form .2n C 1/=20 ; with n 2 Z: This weight occurs in an irreducible summand
of the s -module Vod d : From the classification of the irreducible sl.2; C/-modules, we see that
1
then also occurs as a weight in the irreducible summand, hence in V: Put 0 D 12 : Then
2 0
it follows that gC0 0; hence 0 2 R: Define s0 as above, with 0 in place of : Then
V ./ D V . 0 / D V . 0 /eve n : By the first part of the proof, applied with 0 in place of ; it
follows that V ./ D V . 0 /ev D s0 : This contradicts the fact that gC V ./:
We conclude that Vod d D 0: Hence, V D Vev D s D s and all assertions follow.
Let 2 R: Then by s we denote the B-orthogonal reflection in the hyperplane ker in i t:
Thus, s .H / D H and s D I on ker ; from which one readily deduces that
s .H / D H .H /H .H 2 i t/:
The complex linear extension of s to tC ; also denoted s ; is given by the same formula, for
H 2 tC :
If V is a finite dimensional real linear space, equipped with a positive definite inner product
h ; i; then the map v 7! hv ; i defines a linear isomorphism j W V ! V : We equip V
with the so called dual inner product. This is defined to be the unique inner product that makes
j orthogonal. Thus, if ; 2 V then
1 1 1 1
h ; i D hj ; j i D .j / D .j /:
If A W V ! V is orthogonal, then so is j A j 1 W V ! V : Using the definitions one
readily verifies that j A j 1 D A 1 : In this case we agree to write A for the orthogonal map
A 1 W V ! V : Thus, for 2 V we write A D A 1 :
Following the above convention for V D i t equipped with the positive definite inner product
B; we obtain an orthogonal map s W i t ! i t defined by s D s 1 D s ; for 2
i t : Let H 2 i tI then it follows by application of the above formula for the reflection s that
s ./.H / D .H .H /H / D .H / .H /.H /: From this we see that
s D .H /; . 2 i t /:
Thus, s maps to and is the identity on the hyperplane H0 WD f j .H / D 0g: Since s
is orthogonal it follows that H0 D ? and that s is the orthogonal reflection in the hyperplane
? : The reflection s 2 End.i t / is therefore also given by the formula
h ; i
s D 2 ; . 2 i t /:
h ; i
Comparing this formula with the previous one we see that j.H / D 2=h ; i:
123
Lemma 36.7 Let 2 R: There exists an automorphism ' of gC ; which leaves tC invariant
and has restriction s to this space. The induced endomorphism s of tC leaves R invariant.
Moreover, if 2 R then s ./ 2 Z:
Definition 36.8 The subgroup W D W .g; t/ of GL.i t / generated by the reflections s ; for
2 R; is called the Weyl group of .g; t/:
Proof: By Lemma 36.7, each reflection s leaves R invariant. Hence w.R/ R for each
w 2 W: Since w is injective and R finite, it follows that wjR belongs to the group Sym.R/ of
bijections from R onto itself. Clearly the restriction map r W w 7! wjR ; W ! Sym.R/; is
a group homomorphism. Moreover, since R spans t ; by Lemma 31.9, it follows that ker r is
trivial. Hence #W #Sym.R/ < 1:
124
Let E be a finite dimensional linear space. If 2 E n f0g then by a reflection in we mean
a linear map s W E ! E with s./ D and
E D R ker.s I /:
Lemma 36.10 Let E be a finite dimensional real linear space, and R E a finite subset that
spans E: Then for every 2 R there is at most one reflection s in such that s.R/ D R:
Proof: Let K be the group of A 2 GL.E/ with A.R/ D R: The restriction map r W A 7! AjR is
a group homomorphism from K to the group of bijections of R: Moreover, r has trivial kernel,
since R spans E: It follows that K is a finite group. Hence, there exists an inner product on E
for which K acts by orthogonal transformations (use averaging). If s is any reflection in a non-
zero element of E which preserves R; then it must be an orthogonal transformation, hence the
orthogonal reflection in the hyperplane ? : In particular, there exists at most one such reflection.
Definition 36.11 A (general) root system is a pair .E; R/ consisting of a finite dimensional real
linear space E and a finite subset R E n f0g such that the following conditions are fulfilled.
(a) R spans E:
(b) If 2 R; then R \ R D f ; g:
(c) If 2 R then there exists a (necessarily unique) reflection s in that maps R to itself.
(d) If ; 2 R then s ./ 2 C Z:
According to the results of this section, the pair consisting of E D i t and R D R.g; t/ is a
root system in the sense of the above definition.
For a general root system, the subgroup W of GL.E/; generated by the reflections s ; for
2 R; is called the Weyl group of the root system .E; R/: By the same proof as that of Lemma
36.9, it follows that W is finite. By averaging we see that E may be equipped with a positive
definite inner product h ; i that is W -invariant. It follows that each reflection s ; for 2 R is
orthogonal relative to h ; i: Hence, it is given by the formula
h ; i
s ./ D 2 ; . 2 E/: (55)
h ; i
In terms of the inner product the condition (d) in the definition of root system may therefore be
rephrased as
h ; i
2 2Z .; 2 R/:
h ; i
Two root systems .E; R/ and .E 0 ; R0 / are called isomorphic if there exists a linear isomor-
phism T W E ! E 0 with T .R/ D R0 : If g is a compact semisimple Lie algebra, then it follows
125
from Lemmas 32.2 and 32.3 that the isomorphism class of the root system R.g; t/ is independent
of the choice of the maximal torus t:
We now have the following result, which we state without proof. It reduces the classification
of all compact semisimple Lie algebras to the classification of all root systems.
Theorem 36.12 The map g ! R.gC ; tC / induces a map from (a) the isomorphism classes of real
Lie algebras with negative definite Killing form to (b) the isomorphism classes of root systems.
This map is a bijection.
37 Weyls formulas
We retain the notation of the previous section. In this section we will describe the classification of
of all irreducible representations of the compact semisimple Lie algebra g: Moreover, in terms of
this classification we will state the beautiful character and dimension formulas due to Hermann
Weyl.
The weight lattice D .g; t/ of the pair .g; t/ is defined as the set
D f 2 i t j 8 2 R W s 2 C Zg:
Equip i t with any W -invariant positive definite inner product h ; i: Then from (55) we see
that, alternatively, .g; t/ may be defined as the set of elements 2 i t such that
h ; i
2 2Z for all 2 R:
h ; i
It follows from the definition of root system that the Z-lattice spanned by R is contained in :
The collection of dominant weights (relative to RC ) is defined by
C D f 2 i t j 8 2 RC W s 2 C N. /g:
Thus, C consists of the collection of weights in that are contained in the convex cone
C C D f 2 i t j h ; i 0 for all 2 RC g:
Theorem 37.1 For every 2 C there exists a unique (up to equivalence) irreducible repre-
sentation of g with highest weight :
From this result combined with Theorem 31.23 we obtain the following.
Corollary 37.2 The map 7! induces a bijection from C onto the collection of equiva-
lence classes of irreducible representations of g:
126
Let now G be a compact connected Lie group with algebra g: Let be a finite dimensional ir-
reducible representation of G: Then the associated representation of the Lie algebra is equivalent
to for a unique 2 C : It turns out that in terms of this parametrization, there exist beautiful
formulas for the character and dimension of : The character of is conjugation invariant.
In view of the following result, which we state without proof, it is completely determined by its
restriction to T WD exp.t/:
Proposition 37.3 The group T D exp.t/ is a compact torus in G: Moreover, each element of G
is conjugate to an element of T:
If w 2 W we write .w/ D det.w/ for the determinant of wI since w is orthogonal with
respect to a suitable inner product, we have .w/ D 1: We define the element 2 i t by
1 X
D :
2 C
2R
for all X 2 t for which the denominator is non-zero; these X form an open dense subset (Weyls
character formula). Moreover, the dimension of is given by
Y h C ; i
dim D
C
h ; i
2R
127
38 The classification of root systems
38.1 Cartan integers
In this section we shall study some aspects of the theory of root systems. In particular we shall
describe the first step towards their classification. The starting point of the theory is the definition
of a root system as given in Definition 36.11. In the rest of this section we assume that .E; R/ is
such a root system. The dimension of E is called the rank of the root system.
By the process of averaging over the Weyl group W of the given root system, we select a
W -invariant positive definite inner product h ; i om E: Then, for every 2 R the reflection
s is given by the following formula, for 2 E;
h ; i
s ./ D 2 :
h ; i
s ./ D n ; (56)
see Definition 36.11 (d). These integers are called the Cartan integers for the root system. It
follows from the above representation of the reflection in terms of the inner product that the
Cartan integers are alternatively given by
h ; i
n D 2 : (57)
h ; i
Lemma 38.1 Let ' be an isomorphism from .E; R/ onto a second root system .E 0 ; R0 /: Then,
for all ; 2 R;
(a) ' s D s'./ 'I
(b) n'./ '. / D n :
Proof: It is readily seen that s WD ' s ' 1 W E 0 ! E is a reflection in './: Since s.R0 / D
's ' 1 .'R/ D R0 ; (a) follows. Assertion (b) follows by application of (56).
We shall now discuss the possible values of the Cartan integers. If ; 2 E n f0g; then by
the Cauchy-Schwarz inequality there is a unique ' 2 0; such that
The number ' is called the angle between and (with respect to the given inner product).
Assume that ; 2 R and : Then
jj h ; i
2 cos ' D 2 D n 2 Z:
jj h ; i
128
It follows that
n n D 4 cos2 ' 2 Z:
From this formula we see that the value of ' is independent of the particular choice of W -
invariant inner product on E: By Definition 36.11(b) the roots ; are not proportional, hence
j cos ' j < 1: It follows that
n n 2 f0; 1; 2; 3g:
After renaming we may assume that jj jj: It then follows from (57) that jn j jn j:
By integrality of the Cartan integers we find that either ? or n D 1: This leads to the
following table of possibilities for n and ' :
Lemma 38.2 Let ; 2 R be non-proportional roots with jj jj: Then the following table
contains all possible combinations of values of n ; n and ' : The question mark indicates
that the value involved is undetermined.
j j2 n
n n n n cos ' ' jj2
D n
0 0 0 0 2
1
1 1 1 2 3
1
1 2
1 1 1 2 3
1
1
p
2 2 1 2
2 4
2
1
p 3
2 2 1 2
2 4
2
1
p
3 3 1 2
3 6
3
1
p 5
3 3 1 2
3 6
3
Example 38.3 Let E D R2 ; equipped with p the standard inner product. Let be the first
1 1
standard basis vector .1; 0/; and D . 2 ; 2 3/: Then jj D jj D 1 and ' D 2=3:
p
Moreover, C D . 12 ; 21 3/ has angle =3 with both and : It is easily verified that R D
f; ; . C /g is a root system. Note that n D n D 1: This root system, called A2 ;
is depicted in the illustration following Lemma 38.16. Let r D sC s : Then r is the rotation
over angle 2=3: The reflection s D s is the reflection in the line ? D R.0; 1/: The Weyl
group W equals f1; r; r 1 ; s; s r; s r 1g:
The following lemma will be extremely useful in the further development of the theory.
129
(b) If h ; i < 0 then C 2 R:
Proof: It suffices to establish (a). Then (b) follows by replacing with : Since 2 R is
equivalent to 2 R we may as well assume that jj jj: Then it follows that 0 < n
n ; hence n D 1: In view of (56) this implies that s ./ D : Now use Definition 36.11
to conclude that 2 R:
Given non-proportional roots ; 2 R we define the -string through to be the set
L ./ WD . C Z/ \ R:
The following lemma expresses that root strings have no interruptions and have at most 4 ele-
ments.
Proof: We first establish (a). Write j WD j; for j 2 Z: Assume (a) does not hold. Then
there exist integers k < l such that k ; l 2 R but kC1 ; l 1 R: It follows by application of
Lemma 38.4 that
hk ; i 0 and hl ; i 0:
On the other hand,
contradiction. We conclude that (a) holds. Since 2 L ./; the first assertion of (b) follows.
For the other assertion, we note that s maps L ./ bijectively onto itself. Hence s . C p/ D
C q; from which it follows that n p D q: This establishes (b).
For (c) we note that
D C p is a root. Clearly, L ./ D L .
/ D f
C j j 0
j q pg; so that #L .
/ D q p C 1: It now follows from (b) applied to the pair ;
that
q p D n
; hence n
0 and q p 2 f0; 1; 2; 3g:
130
Definition 38.6 A fundamental system or basis for .E; R/ is a subset S R such that
(a) S is a basis for EI
(b) R NS [ N. S/:
Conditions (a) and (b) of the above definition may be restated as follows. Every root 2 R
admits a unique expression of the form
X
D k ;
2S
Lemma 38.7 Let S be a fundamental system for the root system R: Then for all roots ; 2 S
with one has h ; i 0 (or, equivalently, ' =2).
Proof: Since is a linear combination of the elements of S with both plus and minus signs,
it cannot be a root. It follows from Lemma 38.4 that h ; i 0:
To ensure the existence of fundamental systems, we introduce the notion of a positive system.
By an open half space of E we mean a set of the form E C ./ D fx 2 E j .x/ > 0g; where
is a non-zero element of the dual space E WD HomR .E; R/: Via the given inner product, we
sometimes identify E with E : Accordingly, if
2 E n f0g; we write
E C . / D fx 2 E j hx ; i > 0g:
Definition 38.8 A positive system or choice of positive roots for R is a subset P R with the
following properties.
(a) There exists an open half space containing P:
(b) R P [ . P /:
Lemma 38.9 Let P be a positive system for R. Then S.P / is a fundamental system for R and
P D NS.P / \ R: The map P 7! S.P / is a bijective map from the collection P of positive
systems for R onto the collection S of fundamental systems for R:
131
Proof: Put S D S.P /: Let 2 P: Then either 2 S; or can be written as a sum C
with
;
2 P: Proceeding in this way we see that P NS; hence condition (b) of Definition 38.6
holds. In particular, it follows that S spans E: It remains to be shown that the elements of S are
linearly independent. Let ; be distinct roots in S: By definition of S; neither nor
does belong to P: Hence, R: It follows by application of Lemma 38.4 that h ; i 0:
From the lemma below it now follows that the elements of S are linearly independent. Hence S
is a fundamental system. In the above we established P NS; whence P N. S/ and since
R D P [ . P / it follows that P D NS \ R:
We have shown that the map P 7! S.P / is injective P ! S and will finish the proof
by establishing its surjectivity. For S a fundamental system of R we define RC D RC .S/ D
NS \ R: Since S is a basis for E; the linear functionals h ; i; for 2 S; form a basis for
the dual space E : It follows that there exists a
2 E such that h ;
i > 0 for all 2 S: We
conclude that S; hence RC ; is contained in a half space. From R NS [ . NS/ it follows that
R RC [ . RC /: Hence RC .S/ is a positive system. From S RC .S/ NS it follows that
S.RC .S// S: Both sets of of this inclusion are bases for EI hence, they must be equal. We
conclude that the map R 7! S.P /; P ! S is bijective with inverse S 7! RC .S/:
Lemma 38.10 Let E be a finite dimensional real linear space, equipped with a positive definite
inner product. Let S E be a finite subset contained in a fixed open half space and such that
h ; i 0 for all distinct ; 2 S: Then the elements of S are linearly independent.
P There exists a 2 E such that h ; i > 0 for all 2 S: Let 2 R; for 2 S; be such
Proof:
that 2S PD 0: We define S WD f 2 S j > 0g: Then SC and S are disjoint. We
define WD 2S j j: If one of the sets of summation is empty, the sum is understood to
be zero. The linear relation between the elements of S may now be expressed as C D 0 or
C D :
From the fact that h ; i 0 for all 2 SC and all 2 S it now follows that hC ; C i D
hC ; i 0: Hence,
P hC ; C i D 0 and we conclude that C D D 0: We now observe that
0 D h ; C i D 2SC j jh ; i: Since each of the inner products h ; i is strictly positive,
it follows that SC D ;: Similarly, S D ; and we conclude that D 0 for all 2 S: This
establishes the linear independence.
For each 2 R; the set P WD ker.I s / is called the root hyperplane associated with :
Relative to the given W -invariant inner product, P D ? : We define the set of regular points in
E by
E reg WD E n [2R P : (58)
Since R is a finite set, it is easy to show that E reg is an open dense subset of EI in particular, the
set of regular points is non-empty.
We can now establish the existence of positive systems, hence also of fundamental systems.
For
2 E reg we define
RC . / D R \ E C . / D f 2 R j h ; i > 0g:
132
Lemma 38.11 For every
2 E reg the set RC .
/ is a positive system for R: Moreover, every
positive system arises in this way.
Proof: That RC .
/ is a positive system is immediate from the definitions. Conversely, let P be
a positive system for R; and let S D S.P / be the associated fundamental system. The linear
functionals h ; i; for 2 S; form a basis for the dual space E : Hence there exists a
2 E
such that h
; i > 0 for all 2 S: From P NS it now follows that P RC .
/; hence also
P \ RC .
/ D ;: Since P is a positive system, RC .
/ P [ . P /; so we must have that
P D RC .
/:
Definition 38.12 Let S be a fundamental system for R: The integers n ; for ; 2 S; are
called the Cartan integers associated with S: The square matrix n W S S ! Z; .; / 7! n
is called the Cartan matrix for S:
We will end this section with a result that asserts that every root system is completely de-
termined by the Cartan matrix of a fundamental system. It depends crucially on the following
lemma and Lemma 38.5.
Lemma 38.13 Let S be a fundamental system for R and RC D R \ NS the associated positive
system. If 2 RC n S; then there exists an 2 S such that 2 RC :
P
Proof: Since is not simple, it is of the form
2S k
; k
2 N; with at least two coefficients
non-zero. Thus, if 2 S and 2 R; then at least one of the coefficients of is still
positive, and it follows that 2 RC :
Assume that RC for all 2 S: Then it follows that R for all 2 S:
By Lemma 38.4 this implies that h ; i 0 for all 2 S; hence h ; i 0; hence D 0;
contradiction.
Given any finite set S we write ES for the real linear space with basis S: As a concrete
model we may take the space RS of functions S ! RI here S is embedded in RS by identifying
an element
P 2 S with the function W S ! R given by 7! : If v 2 ES ; we put
v D 2S v : With the above identification, as an element of RS ; the vector v is given by
7! v :
Let E be a real linear space and f W S ! E a map, then f has a unique extension to a linear
map ES ! E; again denoted by f: Moreover, if f W S ! S 0 is a map, then f may be viewed
as a map S ! ES 0 which in turn has a unique linear extension to a map f W ES ! ES 0 :
Theorem 38.14 There exists a map R assigning to every pair consisting of a finite set S and a
function n W S S ! Z a finite subset R.S; n/ ES with the following properties.
(a) If ' W S 0 ! S is a bijection of finite sets, and n W S S ! Z a function, then the induced
map ' W ES 0 ! ES maps R.S 0 ; ' n/ bijectively onto R.S; n/:
(b) If .E; R/ is a root system with fundamental system S and Cartan matrix n W S S ! Z;
then the natural map ES ! E maps R.S; n/ bijectively onto R: In particular, .RS ; R.S; n//
is a root system isomorphic to .E; R/:
133
Remark 38.15 The above result guarantees that the isomorphism class of a root system can be
retrieved from the Cartan matrix of a fundamental system. Later we will see that all fundamental
systems are conjugate under the Weyl group, so that all Cartan matrices of a given root system
are essentially equal, cf. Lemma 38.1.
In the proof of the above result the set R will be defined by means of a recursive algorithm
with input data S; n: This algorithm will provide us with a finite procedure for finding all root
systems of a given rank. Let such a rank r be fixed. Let S be a given set with r elements. Each
root system R of rank r can be realized in the linear space ES ; having the standard basis as
fundamental system.
The possible Cartan matrices run over the finite set of maps S S ! f0; 1; 2; 3g:
For each such map n it can be checked whether or not .ES ; R.S; n// is a root system with
fundamental system S: Condition (b) guarantees that all root systems of rank r are obtained in
this way.
Proof: We shall describe the map R and then show that it satisfies the requirements. Require-
ment (b) is motivational for the definition.
For each 2 S we define the map n W S ! Z by n ./ D n.; /: As said above this map
induces a linear map n W ES ! R: If the linear maps n ; for 2 S; are linearly dependent, we
define R.S; n/ D ; (we need not proceed, since n can impossibly be a Cartan matrix of a root
system). Thus, assume that the n are linearly independent linear functionals.
We consider the semi-lattice D NS ES : Then for each 2 S the map n has integral
values on : We define a height function on in an obvious manner,
X
ht./ D :
2S
Let k be the finite set of 2 with ht./ D k: We put P1 D S and more generally will define
sets Pk k by induction on k:
Let P1 ; : : : ; Pk be given, then PkC1 is defined as the subset of kC1 consisting of elements
that can be expressed in the form C with .; / 2 S Pk satisfying the following conditions.
(ii) jn . C /j 3:
We define P.S; n/ to be the union of the sets Pk ; for k 1 and put R.S; n/ D P.S; n/ [
. P.S; n//:
The set F of 2 ES with n ./ 2 f0; 1; 2; 3g for all 2 S is finite, because the n are
linearly independent functionals. In fact, #F .#S/7 : From the above construction it follows
that R.S; n/ F; hence is finite. In particular, we see that the above inductive definition starts
producing empty sets at some level. In fact, let N be an upper bound for the height function on
F; then P.S; n/ D P1 [ [ PN :
134
From the definition it is readily seen that the map R defined above satisfies condition (a)
of the theorem. We will finish the proof by showing that condition (b) holds. Sssume that
S is a fundamental system for a root system .E; R/: Let RC D R \ NS be the associated
positive system and n W S S ! Z the associated Cartan matrix. The inclusion map S E
induces a linear isomorphism ES ! E via which we shall identify. Then it suffices to show that
R.S; n/ D R: Since n is a genuine Cartan matrix, the functionals n ; for 2 S are linearly
independent. Thus it suffices to show that Pk D R \ k ; for every k 2 N: We will do this by
induction on k: For k D 1 we have R \ k D S D P1 ; and the statement holds. Let k 1 and
assume that Pj D R \ j for all j k: We will show that PkC1 D R \ kC1 :
First, consider an element of PkC1 : It may be written as C with .; / 2 S Pk satisfying
the conditions (i)-(iii). By the inductive hypothesis, 2 RC : Moreover, there exists a smallest
integer p 0 0 such that C p 0 2 RC : By the inductive hypothesis it follows that p 0 D p:
The -string through now takes the form L ./ D f C k j p k qg with q the non-
negative integer determined by p C q D n ./: From condition (iii) it follows that q > 0;
hence C 2 RC : It follows that PkC1 R \ kC1 : For the converse inclusion, consider an
element 1 2 RC of weight k C 1: Since k C 1 2; the root 1 does not belong to S: By Lemma
38.13 there exists a 2 S such that WD 1 2 RC : Clearly, ht./ D k; so 2 Pk by the
inductive hypothesis. We will proceed to show that the pair .; / satisfies conditions (i)-(iii).
This will imply that 1 2 PkC1 ; completing the proof.
Since 1 is a root, ; hence (i). Since n .1 / D n1 ; condition (ii) holds by Lemma
38.2. The -root string through has the form L ./ D f C k j p k qg; with p the
smallest integer such that C p is a root and with q the largest integer such that C q is a
root. We note that p 0 and q 1: By the inductive hypothesis, p is the smallest integer such
that 2 P1 [ [ Pk : Moreover, by Lemma 38.5, n ./ D n D .p C q/ and (iii) follows.
135
Case n D 1. In this case P1 D f; g: There is only one possible element in P2 ; namely
C : Here p D 0 and n ./ D 1 whence p n ./ > 0 and it follows that C 2 P2 :
The possible elements in P3 are . C / C or . C / C : For the first element, Now p D 1
and n . C / D 1; whence 2 C P3 : Similarly, . C / C P3 : It follows that Pj D ;
for j 3: Hence, R D f; ; . C /g is the only possible root system. We leave it to the
reader to check that it is indeed a root system. It is called A2 :
Case n D 2: We have P1 D f; g and P2 D f Cg: The only possible elements in P3
are . C/C and . C/C: For the first of these we have From p D 1 and n . C/ D 0;
so that p n .C g/ > 0 and C 2 2 P3 : For the second element we have From p D 1
and n . C / D 1; whence p n .C / D 0; from which we infer that 2 C P3 : Thus,
P3 D f C 2g:
The possible elements of P4 are . C2/C and . C2/C: For the first element, p D 2
and n . C 2/ D 2; hence C 3 P4 : For the second element, p D 0 and n . C 2/ D 0;
hence 2 C 2 P4 : We conclude that Pj D ; for j 4:
Thus, in the present case the only possible root system is R D f; ; . C /; . C
2/g: Again we leave it to the reader to check that this is a root system. It is called B2 :
Case n D 3: We have P1 D f; g and P2 D f C g: The possible elements of P3 are
C 2 and 2 C : For the first element we have p;C D 1 and n . C / D 1; hence
C 2 2 P3 : For the second we have p;C D 1 and n . C / D 1; hence 2 C P3 :
Thus, P3 D f C 2g:
The possible elements of P4 are C3 and 2C2: For the first element we have p;2C D
2 and n .2 C/ D 1; hence C3 2 P3 : For the second, p;2C D 0 and n .2 C/ D 0;
hence 2 C 2 P4 : Thus, P4 D f C 3g:
The possible elements of P5 are C 3 C and C 3 C : For the first element we have
p D 3 and n . C 3/ D 3; whence C 4 P5 : For the second element we have p D 1
and n . C 3/ D 1; whence 2 C 3 2 P5 ; and we conclude that P5 D f2 C 3g:
The possible elements of P6 are 2 C 3 C and 2 C 3 C : For the first element we
have p D 0 and n .2 C 3/ D 0; and for the second p D 1 and n .2 C 3/ D 1: Hence
Pj D ; for j 6:
We conclude that the only possible root system is R D f; ; C ; 2 C ; 3 C ; 3 C
2g: We leave it to the reader to check that this is indeed a root system, called G2 :
Lemma 38.16 Up to isomorphism, the rank two root systems are completely classified by the
integer n n ; for f; g a fundamental system. The integer takes the values f0; 1; 2; 3g; giving
the root systems A1 A1 ; A2 ; B2 and G2 ; respectively.
136
C
A1 A1 A2
2 C 3
C C 2
C C 2 C 3
B2 G2
137
is an open polyhedral cone. We denote the set of Weyl chambers by C: If C 2 C then for
every 2 R the functional h ; i is nowhere zero on C; hence either everywhere positive or
everywhere negative. We define
RC .C / D f 2 R j h ; i > 0 on C g:
Note that for every
2 C we have RC .C / D RC .
/: Thus, by Lemma 38.11 the set RC .C / is
a positive system for R and every positive system arises in this way.
If C is a Weyl chamber, then by S.C / we denote the collection of simple roots in the positive
system RC .C /: According to Lemma 38.9 this is a fundamental system for R:
Proposition 38.17
(a) The map C 7! RC .C / defines a bijection between the collection of Weyl chambers and
the collection of positive systems for R:
(b) The map C 7! S.C / defines a bijection between the collection of Weyl chambers and the
collection of fundamental systems for R:
(c) Is C is a Weyl chamber, then
Proof: Recall that we denote the collections of Weyl chambers, positive systems and fundamen-
tal systems by C; P and S; respectively.
If P 2 P we define C.P / WD fx 2 E j 8 2 P W hx ; i > 0g; and if S 2 S we
put C.S/ WD fx 2 E j 8 2 S W hx ; i > 0g: With this notation, assertion (c) becomes
C D C.RC .C // D C.S.C // for every C 2 C:
Let S 2 S: Then the set C.S/ is non-empty and convex, hence connected. Since R
NS [ NS; it follows that C.S/ E reg : We conclude that there exists a connected component
C 2 C such that C.S/ C: Every root from R has the same sign on C as on C.S/I hence,
C C.S/: We conclude that C.S/ D C: In particular, S 7! C.S/ maps S into C:
Let P 2 P and let S be the collection of simple roots in P: From S P NS it readily
follows that C.S/ D C.P /: In particular, C.P / 2 C:
From Lemma 38.11 it follows that the map C 7! RC .C / is surjective. If C 2 C then from
the definitions it is obvious that C C.RC .C // C.S.C //: The extreme members in this
chain of inclusions are Weyl chambers, i.e., connected components of E reg ; hence equal. Thus
(c) follows. Moreover, C.RC .C // D C; from which it follows that C 7! RC .C / is injective,
whence (a). Finally, (b) follows from (a) and (c) combined with Lemma 38.9.
The following result gives a useful characterization of the simple roots in terms of the asso-
ciated Weyl chamber.
Lemma 38.18 Let C be an open Weyl chamber. A root 2 R belongs to the associated
fundamental system S.C / if and only if the following two conditions are fulfilled.
(a) h ; i > 0 on C I
138
(b) C \ ? has non-empty interior in ? :
Proof: Put S D S.C / and assume that 2 S: Then (a) follows by definition. From Proposition
38.17 we know that C consists of the points x 2 E with hx ; i > 0 for all 2 S: Since
S is a basis of the linear space E; it is readily seen that C consists of the points x 2 E with
hx ; i 0 for all 2 S: The functionals h ; ij? ; for 2 S n fg; form a basis of ? ; hence
the set CN \ ? contains the non-empty open subset of ? consisting of the points x 2 ? with
hx ; i > 0 for all 2 S n fg: This implies (b).
Conversely, assume that is a root and that (a) and (b) are fulfilled. From (a) it follows that
2 RC .C /: It remains to be shown that is indecomposable. Assume the latter were not true.
Then D C
; for ;
2 RC .C /: From (b) it follows that h ; i 0 and h
; i 0 on an
open subset U of ? On the other hand, h C
; i D 0 on U: It follows that h ; i and h
; i
are zero on U; hence on ? by linearity. From this it follows in turn that ? D ? D
? : Hence
and
are proportional to ; contradiction.
The Weyl group leaves R; hence E reg ; invariant. It follows that W acts on the set of connected
components on E reg ; i.e., on the set C of Weyl chambers. Clearly, W acts on the set of positive
systems and on the set of fundamental systems, and the actions are compatible with the maps
of Proposition 38.17. More precisely, if w 2 W and C 2 C; then RC .wC / D wRC .C / and
S.wC / D wS.C /:
Lemma 38.19 Let RC be a positive system for R and let be an associated simple root. Then
s maps RC n fg onto itself.
P
Proof: Let S be the set of simple roots in RC and let 2 RC ; : Then P D
2S k
;
with k
2 N and k
0 > 0 for at least one
0 different from : Now s ./ D
2S nfg k
C l
for some l 2 Z: Since s is a root, it either belongs to NS or to NS: The latter possibility is
excluded by k
0 > 0: Hence s 2 NS \ R D RC :
If RC is a positive system for R; we define .RC / D to be half the sum of the positive
roots, i.e.,
1 X
D
:
2 C
2R
139
Definition 38.21 Two Weyl chambers C and C 0 are called adjacent if d.C; C 0 / D 1; i.e., the
chambers are separated by precisely one root hyperplane.
Lemma 38.22 Let C; C 0 be Weyl chambers. Then C; C 0 are adjacent if and only if C 0 D s .C /
for some 2 S.C /: If the latter holds, then 2 S.C 0 /:
Lemma 38.23 Let C; C 0 be distinct Weyl chambers. Then there exists a chamber C 00 that is
adjacent to C 0 and such that d.C; C 00 / D d.C; C 0 / 1:
Lemma 38.24 Let C be a Weyl chamber and S D S.C / the associated fundamental system.
Then for every Weyl chamber C 0 C there exists a sequence s1 ; : : : sn of reflections in roots
from S such that C 0 D s1 sn .C /:
Proof: We give the proof by induction on d D d.C; C 0 /: If d D 1; then the result follows
from Lemma 38.22. Thus, let d > 1 and assume the result has been established for C 0 with
d.C; C 0 / < d: By the previous lemma, there exists a chamber C 00 ; adjacent to C 0 and such that
d.C; C 00 / D d.C; C 0 / 1: By Lemma 38.22, C 00 D s .C 0 / for a simple root 2 S.C 0 /:
By the induction hypothesis there exists a w 2 W that can be expressed as a product of
reflections in roots from S.C / such that w.C / D C 00 : Thus, s w.C / D s .C 00 / D C 0 : Moreover,
s w D wsw 1 D ws w 1 ; and since 2 S.C 00 /; it follows that WD w 1 belongs to
S.C / D w 1 S.C 00 /: We conclude that C 0 D ws.C / with w a product of reflections from roots
in S.C / and with s D s ; reflection in a root from S.C /:
Lemma 38.25 Let S be a fundamental system for R: Then every root from R is conjugate to
a root from S by an element of W that can be written as a product of simple reflections, i.e.,
reflections in roots from S:
140
Proof: Let 2 R: There exists a Weyl chamber C such that ? \ C has non-empty interior in
? : By Lemma 38.18 it follows that either or belongs to S.C /: Replacing C by s .C / if
necessary, we may assume that 2 S.C /: Let C C be the unique Weyl chamber with S.C C / D
S: Then there exists a Weyl group element of the form stated such that w 1 .C / D C C : It follows
that w 2 S.C C / D S:
Corollary 38.26 Let S be a fundamental system for R: Then W is already generated by the
associated collection of simple reflections.
Proof: Let W0 be the subgroup of W generated by reflections in roots from S: Let 2 R: Then
by the previous lemma there exists a w 2 W0 such that D w; with 2 S: It follows that
s D ws w 1 2 W0 : Since W is generated by the s ; for 2 R; it follows that W D W0 :
Lemma 38.28 Let 1 ; : : : ; n 2 S be simple roots (possibly with repetitions), and let sj D sj
be the associated simple reflections. Assume that s1 sn .n / is positive relative to S: Then
s1 sn is not a reduced expression. More precisely, there exists a 1 k < n such that
s1 sn D s1 sk 1 skC1 sn 1 :
s1 sn D s1 sk sk sn 1 D s1 sk 1 skC1 sn 1 :
Lemma 38.29 The Weyl group acts simply transitively on the set of Weyl chambers.
Proof: Let C denote the collection of Weyl chambers. The transitivity of the action of W on
C follows from Lemma 38.24. To establish that the action is simple, we must show that for all
C 2 C and w 2 W; wC D C ) w D 1:
Fix C 2 C and let S D S.C / be the associated fundamental system for R: Let w 2 W n f1g:
Then w 1 has a reduced expression of the form w 1 D s1 sn ; with n 1; sj D sj ; j 2
S.C /: From Lemma 38.28 it follows that w 1 n < 0 on C; hence n < 0 on w.C /: It follows
that w.C / C:
141
Remark 38.30 It follows from the above result, combined with Proposition 38.17, that the Weyl
group acts simply transitively on the collection of fundamental systems for R as well as on the
collection of positive systems.
Let S; S 0 be two fundamental systems, and let w be the unique Weyl group element such that
w.S/ D S 0 : Let n W S S ! Z and n0 W S 0 S 0 ! Z be the associated Cartan matrices.
Then it follows from Lemma 38.1 that n0 .w; w/ D n.; / for all ; 2 S; or more briefly,
w n0 D n: Thus, the Cartan matrices are essentially equal.
Let S be a fixed fundamental system for R: From now on we denote the associated positive
system by RC : The elements of S are called the simple roots, those of RC are called the positive
roots. The associated Weyl chamber
E C D fx 2 E j 8 2 RC W h ; xi > 0g
is called the associated positive chamber. Given a root ; we will use the notation > 0 to
indicate that 2 RC I this is equivalent to h ; i > 0 on E C :
We define numbers lS .w/ D l.w/ and nS .w/ D n.w/ for a Weyl group element w 2 W:
Firstly, l.w/; the length of w; is by definition the shortest length of a reduced expression for
w: Secondly, n.W / is the number of positive roots 2 RC such that w is negative, i.e.,
w 2 RC :
Remark 38.31 In general, the numbers lS .w/ and nS .w/ do depend on the particular choice of
fundamental system. This can already be verified for the root system A2 :
Lemma 38.32 For every w 2 W;
n.w/ D l.w/ D d.E C ; w 1
.E C // D d.E C ; w.E C //:
Moreover, any reduced expression for w; relative to S; has length l.w/:
Proof: d.E C ; w 1 .E C // equals the number of positive roots 2 RC such that < 0 on
w 1 .E C /: The latter condition is equivalent with w < 0 on E C or w 2 RC : Thus, n.w/ D
d.E C ; w 1 .E C //: On the other hand, clearly
d.E C ; w 1 .E C // D d.wE C ; ww 1
E C / D d.E C ; wE C /:
It follows from the proof of Lemma 38.24 that any reduced expression has length at most
d.E; wE C /: In particular, l.w/ d.E C ; wE C /:
We will finish the proof by showing that n.w/ l.w/; by induction on l.w/: If l.w/ D 1;
then w is a simple reflection, and the inequality is obvious. Thus, let n > 1 and assume the
estimate has been established for all w with l.w/ < n: Let w 2 W with l.w/ D n: Then w
has a reduced expression of the form w D s1 sn 1 s ; with 2 S.C /: Put v D s1 : : : sn 1 I
this expression must be reduced, hence l.v/ < n and it follows that n.v/ n 1 by the
inductive hypothesis. On the other hand, from Lemma 38.28 it follows that w 2 RC ; hence
WD v > 0: The root belongs to S.vE C /; hence RC .wE C / D RC .s vE C / D RC .vE C / n
fg [ f g: It follows that RC n RC .wE C / is the disjoint union of RC n RC .vE C / and fg:
Hence n.w/ D d.E C ; wE C / D d.E C ; vE C/ C 1 D n.v/ C 1 l.v/ C 1 l.w/:
142
38.5 Dynkin diagrams
Let .E; R/ be a root system, S a fundamental system for R: The Coxeter graph attached to S is
defined as follows. The vertices of the graph are in bijective correspondence with the roots of SI
two vertices ; are connected by n n edges. Thus, every pair is connected by 0; 1; 2 or 3
edges, see the table in Lemma 38.2.
The Dynkin diagram of S consists of the Coxeter graph together with the symbol > or <
attached to each multiple edge, pointing towards the shorter root. From Lemma 38.16 it fol-
lows that (up to isomorphism) the Dynkin diagrams of the rank-2 root systems are given by the
following list:
A1 A1 A2
B2 G2
It follows from Remark 38.30 that the Dynkin diagrams for two different choices of fundamental
systems for R are isomorphic (in an obvious sense). We may thus speak of the Dynkin diagram
of a root system. The following result expresses that the classification of root systems amounts
to describing the list of all possible Dynkin diagrams.
Theorem 38.33 Let R1 ; R2 be two root systems. If the Dynkin diagrams associated with R1
and R2 are isomorphic, then R1 and R2 are isomorphic as well.
Proof: Let S1 and S2 be fundamental systems for R1 and R2 ; respectively. It follows from
Lemma 38.2 that the Cartan matrices n1 and n2 of S1 and S2 are completely determined by their
Dynkin diagrams. An isomorphism between these Dynkin diagrams gives rise to a bijection
' W S1 ! S2 such that, n1 D ' n2 : By Theorem 38.14 it follows that R1 and R2 are isomorphic.
Remark 38.34 It follows from the above result combined with Theorem 36.12 that the (isomor-
phism classes of) Dynkin diagrams are in bijective correspondence with the isomorphism classes
of semisimple compact Lie algebras.
Let S be a fundamental system. The decomposition of its Dynkin diagram D into connected
components Dj ; .1 j p/; determines a decomposition of S into a disjoint union of subsets
Sj ; .1 j p/: Here Sj consists of the roots labelling the vertices in Dj : The decomposition
of S is uniquely determined by the conditions that Si ? Sj if i j; and that every Sj cannot be
143
written as a disjoint union of proper subsets Sj1 ; Sj 2 with Sj1 ? Sj 2 : We will investigate what
this means for the root system R:
If .Ej ; Rj /; with j D 1; 2; are two root systems, we define their direct sum .E; R/ as
follows. First, E WD E1 E2 : Via the natural embeddings Ej ! E; the sets R1 and R2 may be
viewed as subsets of EI accordingly we define R to be their union. If 2 R1 ; the map s I
is a reflection in .; 0/ preserving R: By a similar remark for R2 ; we see that R is a root system.
Moreover, for all 2 R1 and 2 R2 ; n D 0: From this we see that E1 ? E2 for every
W -invariant inner product on E: Every reflection preserves both R1 and R2 ; hence E1 and E2
are invariant subspaces for the Weyl group. Moreover, the maps v 7! v I and w 7! I w
define embeddings W1 ,! W and W2 ,! W via which we shall identify. Accordingly we have
W D W1 W2 : Similar remarks hold for the direct sum of finitely many root systems.
Definition 38.35 A root system .E; R/ is called reducible if R is the union of two non-empty
subsets R1 and R2 such that E D span.R1 / span.R2 /: It is called irreducible if it is not
reducible.
The following result expresses that every root system allows a decomposition as a direct sum
of irreducibles, which is essentially unique.
Proposition 38.36 Let .E; R/ be a root system. Then there exist finitely many linear subspaces
Ej ; 1 j n; such that Rj WD Ej \ R is an irreducible root system for every j; and such that
R D [j Rj : The Ej are uniquely determined up to order.
If Sj is a fundamental system of Rj ; for j D 1 n; then S D S1 [ [ Sn is a fundamental
system for R: Every fundamental system for R arises in this way.
If Pj is a positive system of Pj ; for j D 1 n; then P D P1 [ [ Pn is a positive system
for R: Every positive system of R arises in this way.
Proof: From the definition of irreducibility, it follows that .E; R/ has a decomposition as stated.
We will establish its uniqueness at the end of the proof.
If the Sj are fundamental systems as stated, then it is readily checked from the definition that
their union S is a fundamental system for R: Let Pj be positive systems as stated, then again
from the definition it is readily verified that their union P is a positive system for R:
Conversely, let P be a positive system for R: Then it is readily verified that every set Pj WD
P \ Rj is a positive system for Rj : Moreover, let S be a fundamental system for R: Since R is
the disjoint union of the sets Rj ; it follows that S is the disjoint union of the sets Sj WD S \ Rj :
Each Sj is linearly independent, hence for dimensional reasons a basis of Ej : Now Rj .NS [
. NS// and Rj RSj : By linear independence this implies that Rj NSj [. NSj / for every
j: Hence every Sj is a fundamental system.
We now turn to uniqueness of the decomposition as stated. Let E D 1j m Ej0 be a de-
composition with similar properties. Fix a fundamental system Sj0 for Rj0 D R \ Ej0 ; for every
j: The union S 0 is a fundamental system for R hence of the form S D S1 [ [ Sn ; with Sj a
fundamental system for Rj ; for each j: It follows that S10 is the disjoint union of the sets S10 \ Sj ;
1 j n: Hence E10 is the direct sum of the spaces E10 \ Ej and R10 is the union of the sets
R10 \ Rj D R10 \ Ej : From the irreducibility of E10 it follows that there exists a unique j such
that E10 D Ej : The other components may be treated similarly.
144
In view of the above result we may now call the uniquely determined .Ej ; Rj / the irreducible
components of the root system .E; R/:
Lemma 38.37
(a) Let R be a root system. Then the Dynkin diagram of R is the disjoint union of the Dynkin
diagrams of the irreducible components of R:
(b) A root system is irreducible if and only if the associated Dynkin diagram is connected.
Proof: Let .E; R/ be an root system, with irreducible components .Ej ; Rj /: Select a fundamen-
tal system Sj for each Rj and let S be their union. The inclusion Sj S induces an inclusion of
Dj ,! D via which we may identify. For distinct indices i; j we have n D 0 for all 2 Si ;
2 Sj : Hence no vertex of Di is connected with any vertex of Dj : It follows that D is the
disjoint union of the Dj ; and (a) follows.
We turn to (b). If R is reducible, then by (a), the associated Dynkin diagram is not connected.
Conversely, assume that the Dynkin diagram of R is not connected. Then it may be written as the
disjoint union of two non-empty diagrams D1 and D2 : Fix a fundamental system S of R: Then
S decomposes into a disjoint union of two non-empty subsets S1 and S2 such that the elements
of Sj label the vertices of Dj : It follows that for all 2 S1 and all 2 S2 ; n D 0: Put
Ej D span.Sj /; then it follows that for each 2 S the reflection s leaves the decomposition
E D E1 E2 invariant. Hence, the Weyl group W of R leaves the decomposition invariant. Let
2 R; then there exists a w 2 W such that w 2 S D S1 [ S2 : It follows that lies either in
E1 or in E2 : Hence R D R1 [ R2 with Rj D Ej \ R; and we see that R is reducible.
The following result relates the notion of irreducibiliy of a root system with decomposability
of a semisimple Lie algebra.
Proposition 38.38 Let g be a compact semisimple Lie algebra with Dynkin diagram D: Let
D D D1 [ : : : [ Dn be the decomposition of D into its connected components. Then every Dj
is the Dynkin diagram of a compact simple Lie algebra gj : Moreover,
g ' g1 gn :
Remark 38.39 Note that in view of Lemma 35.10 the above result implies that the connected
components of D are in bijective correspondence with the simple ideals of g:
Proof: Let g D j hj be the decomposition of g into its simple ideals. For each j we fix a
maximal torus tj hj : Then t WD t1 tn is a maximal torus in g (use that hi commutes
with hj for every i j ). Via the direct sum decomposition of t; we view tj as the linear
subspace of elements of t that vanish on tk for every k j: Accordingly, t D t1 tn ;
and a similar decomposition of the complexification. Let Rj be the root system of tj in hj : Since
gC is the direct sum of tC and the root spaces gC ; for 2 R1 [ [ Rn ; it follows that the
root system R of t in g equals the disjoint union of the Rj : Hence, R is the direct sum of the Rj :
145
The Dynkin diagram of R is the disjoint union of the Dynkin diagrams of the Rj : The proof will
be finished if we can show that the Dynkin diagram of Rj ; is connected, for each j: By Lemma
38.37 this is equivalent to the assertion that each Rj is irreducible.
Thus, we may assume g is simple, t a maximal torus in g; and then we must show that
R D R.g; t/ is irreducible. Assume not. Then we may decompose R as the disjoint union of
two non-empty subsets R1 and R2 whose spans have zero intersection. Put E D i t ; and for
j D 1; 2; define Ej D span.Rj /: Then E D E1 E2 : Let
Theorem 38.40 The following is a list of all connected Dynkin diagrams of root systems. These
diagrams are in bijective correspondence with the (isomorphism classes of) the simple compact
Lie algebras.
146
An W ::: n1 SU.n C 1/
G2 W
F4 W
E6 W
E7 W
E8 W
Acknowledgement: I warmly thank Lotte Hollands for providing me with LATEX files for these
and all other pictures in the lecture notes.
147
Index
abelian group, 4 Coxeter graph, 143
action of a group, 43 cyclic vector, 111
adjacent Weyl chambers, 140
adjoint representation, of G, 18 densities, bundle of, 63
angle, between roots, 128 density, invariant, 64
anti-symmetry, of Lie bracket, 20 density, on a linear space, 62
arcwise connected, 8 density, on a manifold, 63
associativity, 4 derivation, 115
automorphism, of a Lie group, 7 direct sum of representations, 83
dominant, 126
Banach space, 69 dual of a representation, 82
Banach-Steinhaus theorem, 71 Dynkin diagram, 143
barrelled space, 71
base space, of principal bundle, 48 equivalence class, 5
basis, of root system, 131 equivalence relation, 5
equivalent representations, 74
Cartan integer, 128 equivariant map, 43, 74
Cartan integers, 133 exponential map, 16
Cartan matrix, 133
center of a group, 5 fiber, of a map, 5
center, of a Lie algebra, 116 finite dimensional representation, 69
character of a representation, 80 frame bundle, 48
character, multiplicative, 94 free action, 50
choice of positive roots, 131 fundamental system, 131
class function, 93 G-space, 43
closed subgroup, 12 group, 4
commutative group, 4 group of automorphisms, 114
commutative Lie algebra, 22 group of interior automorphisms, 115
commuting elements, of the Lie algebra, 22
compact Lie algebra, 116 Haar measure, 66
complex Hilbert space, 72 Haar measure, normalized, 66
complexification, of a Lie algebra, 100 half space, 131
component of the identity, 23 height of a root, 131
conjugation, 5 Hermitian inner product, 72
connected, 8 highest weight, 112
continuous action, 43 highest weight vector, 110
continuous representation, 69 Hilbert space, 69
contragredient representation, 82 homogeneous, 55
coset space, 6 homomorphism, of groups, 4
coset, left, 6 homomorphism, of Lie groups, 7
148
ideal, 60 normal subgroup, 6
image, of a homomorphism, 4 normalized Haar measure, 66
indecomposable root, 131
induced infinitesimal representation, 106 one parameter subgroup, 18
integral curve, 16 open half space, 131
integral operator, 89 open subgoup, 24
integral, of a density, 64 orbit space, 44
intertwining map, 43, 74 orbits, for group action, 44
invariance, of Killing form, 116 orthogonal group, 13
invariant density, 64 partition, 5
invariant subspace, 72 Peter-Weyl theorem, 86
inverse function theorem, 17 positive density, 63
irreducible representation, 72 positive density, on a manifold, 63
isomorphic, 4 positive root, 110
isomorphism, 4 positive system, 131
isomorphism of Lie groups, 7 primitive vector of an sl(2)-module, 103
isomorphism of root systems, 125 principal fiber bundle, 48
Jacobi identity, 21 product density, 88
proper action, 49
kernel, of a group homomorphism, 4 proper map between topological spaces, 49
kernel, of an integral operator, 89
killing form, 115 Radon measure, 65
rank, of a root system, 128
Lebesgue measure, 63 real symplectic group, 13
left action, 43 reducible root system, 144
left invariant vector field, 15 reflection, 125
left regular representation, 70 regular element, 113
left translation, 5 relation, 5
Lie algebra, 21 representation, of a Lie algebra, 70
Lie algebra homomorphism, 21 representative functions, 85
Lie subgroup, 28 right action, 43
local trivialization, of principal bundle, 48 right regular representation, 70
locally convex space, 69 right translation, 5
Lorentz group, 13 root space, 107
matrix coefficient, of representation, 73 root space decompostion, 107
maximal torus, 106 root system, general, 125
module, for a Lie algebra, 70 roots, 107
module, of a Lie group, 71 Schur orthogonality, 79
monomorphism, 4 Schur orthogonality relations, 79
multiplicative character, 94 Schurs lemma, 75
multiplicity, of an irreducible representation, 84 semisimple Lie algebra, 119
neutral element, 4 sesquilinear form, 72
149
simple ideal, 119
simple Lie algebra, 119
simple root, 131
slice, 50
smooth action, 45
special linear group, 10
special orthogonal group, 13
special unitary group, 13
spectral theorem, 87
standard sl(2)-triple, 102
structure group, of principal bundle, 48
subalgebra of a Lie algebra, 30
subgroup, 4
submersion theorem, 11
substitution of variables, for density, 64
symplectic form, 13
symplectic group, compact form, 15
symplectic group, complex form, 15
system of positive roots, 110
vector field, 15
weight, 105
weight lattice, 126
weight space, 105
Weyl chamber, 109, 137
Weyl group, of a compact algebra, 124
Weyl group, of root system, 125
Weyls character formula, 127
Weyls dimension formula, 127
150