Journal of Algebra: Miriam Cohen, Sara Westreich
Journal of Algebra: Miriam Cohen, Sara Westreich
Journal of Algebra: Miriam Cohen, Sara Westreich
Journal of Algebra
www.elsevier.com/locate/jalgebra
a r t i c l e i n f o a b s t r a c t
Introduction
The representation and character theory of semisimple Hopf algebras over an algebraically closed
field of characteristic zero has been developed since the 70s, in many cases analogously to the classi-
cal theory of finite groups. A comprehensive discussion of this theory is given by Montgomery in [19].
In this paper we survey some known results and prove new ones about characters and structure con-
stants for both semisimple and non-semisimple symmetric Hopf algebras from the point of view of
symmetric algebras. Symmetric algebras are abundant. Finite group algebras over any field, finite-
dimensional semisimple algebras, and many quantum groups.
* Corresponding author.
E-mail addresses: mia@math.bgu.ac.il (M. Cohen), swestric@mail.biu.ac.il (S. Westreich).
0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jalgebra.2010.07.003
3220 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
One of the best known structure constants are the so-called “fusion rules” that describe the prod-
uct of simple objects in fusion categories. A particular example is the product of irreducible characters
for semisimple factorizable Hopf algebras. A famous formula which is associated to them is the “Ver-
linde formula”. In [6] we have given a purely algebraic proof for the Verlinde formula using the
so-called quantum Fourier transform. Another important instance of structure constants exists for fi-
nite groups and the way their class sums multiply. In [8] we have found a generalization of conjugacy
classes and class sums for semisimple Hopf algebras H whose character algebra is commutative, and
a formula for their associated structure constants. When H is factorizable these constants turn out to
equal the fusion rules up to rational scalar multiples. In this paper we further study the conjugacy
classes and prove a connection between them and the commutator subspace of H . This connection
boils down to a known connection for finite group algebras.
Much less is known about representations and characters of general finite-dimensional Hopf al-
gebras H . While the building blocks in the representation theory of finite-dimensional semisimple
algebras are the irreducible modules, in the non-semisimple case, a major role is also played by
indecomposable projective modules. There is a reflection of these differences in H ∗ . As in [8] we con-
sider a chain of subspaces of the cocommutative elements of H ∗ , Coc( H ∗ ), namely P ⊂ R ⊂ Coc( H ∗ ),
where P is the k-span of characters of all projective H -modules and R is the k-span of all irre-
ducible characters (sometimes called the character algebra). We relate them to a chain of subspaces
in H via a sort of Fourier transform. The latter is a known chain for any symmetric algebra A:
Higman( A ) ⊂ Reynolds( A ) ⊂ Z ( A ). The collapse of the chain to Coc( H ∗ ) (or respectively to Z ( H ))
is in fact equivalent to the semisimplicity of H .
We mainly focus on non-semisimple factorizable ribbon Hopf algebras. The basic examples of such
Hopf algebras are some small quantum groups and the Drinfeld double of certain finite-dimensional
Hopf algebras. While semisimple factorizable Hopf algebras are always ribbon, this is not true in the
non-semisimple case (e.g. the Drinfeld double which is always factorizable but [15] prove a necessary
and sufficient condition for it to be ribbon). Being a ribbon Hopf algebra means that there exists a
specific group-like element G which makes Rep( H ) into a ribbon category. In particular, if V is a
representation of H then the canonical isomorphism between V and V ∗∗ as vector spaces is not an
H -module map, this can be remedied by defining an isomorphism δ V between V and V ∗∗ by
δ V ( v ), v ∗ = v ∗ , G · v ,
for all v ∈ V , v ∗ ∈ V ∗ . This isomorphism gives rise to the notions of quantum trace and quantum
dimension. We study the quantum trace and quantum dimension of elements in the above mentioned
chains.
The quantum dimension is an essential ingredient in generalized fusion rules and other structure
constants which exist in the semisimple case. There are two major differences though which have to
be taken into account. While in the semisimple case G = 1 and hence the quantum trace and quantum
dimension equal the usual ones and so the quantum dimension of any module is different from 0,
in the non-semisimple case there may exist irreducible modules whose quantum dimension is 0. For
instance, for U q (sl2 ) the quantum dimension of the (unique) projective irreducible module is 0 while
it is different from 0 for all other irreducible modules. The second difference is that in the semisimple
case the set of irreducible characters of H is a basis for Coc( H ∗ ) while in the non-semisimple case it
is not.
We focus on R which encodes information about tensor products of H -modules and their decom-
positions. Our methods are reminiscent of the methods employed when analyzing the family of tilting
modules associated with certain quantum groups that give rise to fusion categories (see e.g. [4, 11.3
and 15.3]). They are also reminiscent of “splitting modules” for certain Drinfeld doubles discussed
in [11]. We distinguish between “well-behaved” and “poorly-behaved” modules according to their be-
havior on a subalgebra of H , in particular the vanishing of their quantum dimension. We prove that R
contains an ideal T which contains all the “poorly-behaved” elements. We show that P is contained
in T and prove that a certain matrix defined by the values of the Drinfeld map on well-behaved irre-
ducible characters diagonalizes the appropriate fusion rules on R/T . If the factorizable Hopf algebra
is semisimple then all modules are “well behaved” and we just retrieve the usual Verlinde formula.
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3221
We also define analogues of class sums and conjugacy classes for factorizable ribbon Hopf alge-
bras. This definition is a natural extension of class sums and conjugacy classes for groups and for
semisimple quasitriangular Hopf algebras.
The paper is organized as follows. In Section 2 we consider semisimple Hopf algebras H , whose
character algebra is commutative. We review the notions of class sums C i and conjugacy classes Ci
introduced in [8]. When ( H , R ) is a semisimple factorizable Hopf algebra, we review the intimate
relation between the structure constants associated with the product of class sums and the fusion
rules, pointing out relations which will be used more generally in Section 3. Using ideas from the
theory of symmetric algebras we prove:
Theorem 2.7. Let H be a semisimple Hopf algebras and assume Coc( H ∗ ) is commutative. Let { F i } be the set of
primitive idempotents of Coc( H ∗ ). Define the conjugacy classes Ci by Ci = Λ F i H ∗ . Then the commutator
subspace K satisfies
K= (Ci ∩ ker ε ).
We also prove in Corollaries 2.6 and 2.9 that Ci and K are stable under the adjoint action and
hence that Λad ˙ K = 0, where Λ is a nonzero integral of H .
In Section 3 we focus on non-semisimple factorizable ribbon Hopf algebras. In this situation the
specific group-like element G induces S −2 . Shifting the Drinfeld map f Q by G, denoted here by fQ ,
allows us to generalize results from the semisimple case. We start by defining analogues of class
sums C i and conjugacy class Ci , motivated by the observations in Section 2. Let {χ1 , . . . , χn } be the
set of irreducible characters of H . We prove:
Theorem 3.8. Let ( H , R , v ) be a factorizable ribbon Hopf algebra. For any irreducible character χi set
Ci =
f Q (χi ) H ∗ .
Then:
A fundamental property of conjugacy classes in the semisimple case is their connection to K , the
commutator subspace of H . We show:
In order to find an analogue to the diagonalizing matrix of the fusion rule we must “fine tune” the
analysis. Instead of the usual dimensions of modules V we consider the more appropriate quantum
dimensions χ V , G . For an irreducible χi we denote for simplicity its quantum dimension by qdi . We
consider the set { F 1 , . . . , F s } of all primitive idempotents of R and define a partition on {1, . . . , n} by
M j = k χk
f Q ( F j ) = χk .
T = c ∈ R c , G
f Q (R) = 0
and show:
3222 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
s ji = χi , Gf Q (χ j ) qd −j 1 , s̃ ji = χi , Gf Q (χ j ) .
We prove:
Theorem 3.14. Let ( H , R , v ) be a factorizable ribbon Hopf algebra over an algebraically closed field of char-
acteristic zero. Let (si j ) and (s̃i j ) be defined as above. Then both matrices are invertible and the matrix (s̃i j ) is
symmetric.
We prove also a variation of the Verlinde formula for the appropriate fusion rules. For 1
i , j , l m set
m̃ijl = i
mkl qd k ,
k∈M j
n
where χi χl = i
k=1 mkl χk . We define the matrix Mi as:
Mi jl
= qd −j 1m̃ijl .
Then:
Theorem 3.15. Let ( H , R , v ) be a factorizable ribbon Hopf algebra over an algebraically closed field of char-
acteristic zero. Let T , M j , { F j }, {χ j }, m̃ijl and Mi be as above. Let A = (si j ) and Di = diag{si1 , si2 , . . . , sim }.
Then for each 1 i m:
D i = A −1 M i A .
m
m̃ijl = qd j sti stl u jt ,
t =1
where (u i j ) = A −1 .
1. Preliminaries
Throughout, the base field k is assumed to be algebraically closed of characteristic zero. All algebras
and representations are assumed to be finite-dimensional over k.
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3223
Let A be an algebra and let and the following right and left actions of A on A ∗ :
p a, a = p , aa , a p , a = p , a a .
The elements of Coc( A ∗ ) are called central forms for A. Let J ( A ) (or J when no ambiguity arises)
be the Jacobson radical of A and let A = A / J ( A ). Let { V 1 , . . . , V n } be a full set of non-isomorphic
irreducible left A-modules and {e 1 , . . . , en } the corresponding orthogonal primitive idempotents. That
is, V i ∼
= Ae i . For any primitive idempotent e there exists a unique central primitive idempotent E so
that Ee = e. We say in this case that e belongs to the block A E.
For each finite-dimensional A-module V denote its structure map A → Endk ( V ) by ρ V . Then the
character χ V of V is defined by
χ V , a = Trace ρ V (a) .
χi , e j = δi j . (1)
Set
Proposition 1.1. Let A be a finite-dimensional algebra over an algebraically closed field k. Then the following
are equivalent for x ∈ A ∗ :
(i) x ∈ R( A ) .
(ii) x ∈ Coc( A ∗ ) and x, J ( A ) = 0.
(iii) x ∈ Coc( A ∗ ) and x J ( A ) = 0.
(iv) x ∈ R( A / J ( A )) = spk {χi } where {χi } is
the set of irreducible characters of A (and thus of A / J ( A )).
Moreover, any character χ satisfies χ = ni χi where ni is a non-negative integer for all i.
P = { pa | a ∈ A }. (3)
Proposition 1.2. (See [1, Prop. 4.7].) Let A be a finite-dimensional algebra and let P be defined as in (3).
projective A-module P , with an A-coordinate system {xi , f i }, the character χ P
For each finitely generated
equals pa , where a = f i (xi ). Hence P ⊂ R. Moreover, P is spanned over k by the set { p e }, e a primitive
idempotent of A.
3224 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
Remark 1.3. For a fixed non-idempotent a, the correspondence b
→ l(b) ◦ r (a) : A → A is not multi-
plicative, while for an idempotent e it is. In this case, ρ (b) = l(b) ◦ r (e ) is a representation of A. In
fact, ρ is the representation given by left multiplication on the module Ae. The A-coordinate system
is {e , Id} and p e = χ Ae .
˙a=
had h1 aS (h2 ),
h · (v ⊗ w ) = h1 · v ⊗ h2 · w ,
for all v ∈ V , w ∈ W , h ∈ H . Since the product in H ∗ is a convolution product, it follows that the
character of V ⊗ W satisfies
χ V ⊗W = χ V χW .
We refer to the surveys in [3,13] and describe some properties of such algebras which play a role
in the sequel.
(i) There exists an associative bilinear form β : A ⊗ A → k which is non-degenerate. The form β is
given by β(a, b) = φ(a), b.
(ii) There exists t ∈ A ∗ so that the map φ given by φ(a) = (t a) is a right A-modules isomorphism.
Equivalently, a
→ (a t ) is a left A-modules isomorphism. t is defined by t = φ(1) and A ∗ is a
free A-module with a basis t. Conversely, given t, then the A-module map φ is determined by
φ(1) = t.
(iii) There exist t ∈ A ∗ , ai , b i ∈ A, i = 1, . . . , n, such that for all x ∈ A,
x= ai t , b i x. (4)
We say that {ai , b i } form dual bases for the form β and call ai ⊗ b i the Casimir element
of ( A , t ).
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3225
for all a ∈ A. It is well known that Im(τ ) is an ideal in Z . We call it the Higman ideal of A and
denote it by Hig. The element
τ (1) = ai b i
t , ab = t , ba,
for all a, b ∈ A, then t is a central form and ( A , t ) is called a symmetric algebra. In this case the
Casimir element satisfies
ai ⊗ b i = b i ⊗ ai .
The following are basic examples of symmetric algebras over any field k:
Example 1.6. (i) The group algebra kG, where k is a field and G is a finite group is a motivating
example for both Hopf algebras and symmetric algebras. The set { g } g ∈G is a (standard) k-basis for kG,
and as is well known, its k-dual of (kG )∗ is also a Hopf algebra with a dual basis {π g } g ∈G , where
π g is the projection into the g-component. The central form t is given by π1 , the projection onto k1.
Dual bases for this form are { g , g −1 } g ∈G and the central Casimir element is |G | · 1.
(ii) The simple algebra M n (k). The central form t is the usual trace of a matrix and dual bases
are {e i j , e ji } where e i j are the matrix units. The central Casimir element is nI . Similarly, any finite-
dimensional semisimple algebra is a symmetric algebra [12].
(iii) Let H be a finite-dimensional unimodular Hopf algebra. Assume S 2 is an inner automorphism
induced by l. It was proved in [20] that H is symmetric. In this case the central form t is given by
λ l and corresponding dual bases are {Λ1l−1 , S Λ2 }. Hence
Hig = Λ1 H S Λ2 = Λad˙ H ,
where ad
˙ is the left adjoint action of H on itself (see e.g. [7]).
For any symmetric algebra ( A , t ) the Higman ideal reflects the characters of projective A-modules
as follows:
Proposition 1.7. Let ( A , t ) be a symmetric algebra with a corresponding A-module isomorphism φ . Then for
all a ∈ A,
φ τ (a) = pa .
where Z 0 = E Z E, where E ranges over the set of blocks of A which are simple k-algebra, and the
Reynolds ideal is defined as:
Observe that any idempotent E in Rey, satisfies E J ( A ) = 0 and thus E A is a simple algebra and
E ∈ Z 0 . It follows that
Rey = Z 0 ⊕ N , N ⊂ J ( A ). (7)
Set
Proposition 1.8. Let ( A , t ) be a symmetric algebra. Then the chain (6) corresponds to the following chain
in A ∗ ,
C 0 ⊂ P ⊂ R ⊂ Coc A ∗ (8)
by
C 0 = φ( Z 0 ), P = φ(Hig), R = φ(Rey), Coc A ∗ = φ( Z ).
Remark 1.9. It was proved in [7, Cor. 2.3] that any of the inclusions in (6) or (8) is an equality if and
only if A is semisimple.
Let ( A , t ) be a symmetric algebra with corresponding symmetric form β . For each subspace V
V ⊥ = a ∈ A β(a, V ) = 0 .
As a corollary we obtain:
(i) A = Z ⊕ Z ⊥ = K ⊕ K ⊥ = K ⊕ Z = Ker τ ⊕ Im τ .
(ii) τ (1) is an invertible element of A and an element x ∈ Ker τ can be expressed explicitly as an element in K
as follows:
x = x − τ (1)−1 τ (x) = τ (1)−1 b i ai x − ai xb i .
Proof. (i) If A is semisimple then Coc( A ∗ ) and Z are dual vector spaces by χi , E j = χi , 1δi j , where
{ E j } are the primitive central idempotents of A so that e i belongs to the block E i . This implies by
Lemma 1.11 that K ∩ Z = 0. Also, since Hig = Z we have K = ker τ and Z = Im τ .
(ii) This follows from the last equality above. Explicitly, if τ ( z) = 0 then z ∈ Ker τ ∩ Z = 0. But
τ (z) = τ (1)z, hence τ (1) is a nonzero divisor in Z , which is a direct sum of fields, and thus τ (1) is
invertible. 2
2. Structure constants for semisimple Hopf algebras H with commutative character algebras
We denote by i ∗ , j ∗ the indexes relating s(χi ), s( F j ) respectively. Applying s to both sides of (9)
yields
αi j = αi ∗ j ∗ .
where mijs ∈ N. These are the so-called “fusion rules”. Since { F i } are orthogonal idempotents, (9) im-
plies
χi F j = αi j F j (11)
Theorem 2.1. (See [8, Th. 3.1].) Let H be a semisimple Hopf algebra over an algebraically closed field of char-
acteristic zero so that Coc( H ∗ ) is commutative. Let A = (αi j ) be the change of basis matrix from the set of
primitive idempotents of Coc( H ∗ ), { F j }1 j n to the set of irreducible characters {χi }1i n . Then:
αik α jk αs∗ k
mijs = .
nk
k
In [8] we have defined an analogue of conjugacy classes and class sums for Hopf algebras. It
reduces to the usual definition when applied to finite group. Let H = kG, G a finite group. Then
Λ = |G1 | g ∈G g and |G | = dim( H ) = d. Moreover, F i = g ∈Ci π g is a primitive central idempotent.
The conjugacy classes and the class sums for G are defined respectively by
C i = g −1 x i g , g ∈ G and C i = g.
g ∈Ci
Thus kCi is the right coideal (in fact the subcoalgebra) of kG generated by the central element C i . We
have C i = Λ dF i , and |Ci | = dim( F i H ∗ ). This motivates the following definition:
Definition 2.2. Let H be a semisimple Hopf algebra so that Coc( H ∗ ) is commutative, and let Λ be an
integral so that ε (Λ) = 1. Let { F i } be the set of central primitive idempotents of Coc( H ∗ ). Define the
i-th class sum C i by
C i = Λ dF i (12)
Ci C j = c ijt C t . (13)
t
dim( H ∗ )
Call {c ijt } the structure constants of the product of the conjugacy classes. Let ni = dim( H ∗ F i )
which
is known to be an integer by [14,22]. Let
α α2i
Di = d
ni
1 d2
, . . . , αdnin }, where {α ji } are as in (9). Let
diag{ d1i ,
i ∈ Matn (k) be the matrix of l(C i ) with respect to the basis {C j }. Let
M A = (
αi j ) be the change of basis
matrix from {C i } to { E j }. Then by [8, (18)]
d
αi j = α ji .
ni d j
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3229
We prove
Theorem 2.3. (See [8, Th. 3.8].) Let H be a semisimple Hopf algebra over an algebraically closed field of char-
acteristic zero and assume Coc( H ∗ ) is commutative. Let { F i } be a set of primitive idempotents of Coc( H ∗ ) and
let { E i } be a set of primitive idempotents in Z . Let C i = dF i Λ. Then:
(i)
A diagonalizes the structure constants. That is, for each i,
Di = i
A −1 M A.
(ii) C i E j =
αi j E j and thus { E j } is a basis of eigenvectors for l(C i ) for all i.
(iii) Let kt ). Then β
A −1 = (β kt = dk
αk∗ t .
d
(iv) C i C j = i
t c jt C t where c ijt satisfies
In the following example we show that when H = kG, G a finite group, then the formula in
Theorem 2.3(iv) is just the formula for the structure constants recovered from the character table.
|G |
Example 2.4. Let H = kG, G a finite group. Since |Ci | = dim( H ∗ F i ), we have ni = |C | = |C G ( g i )|, where
i
g i is an element of Ci .
j
For a group G, its character table (see [9, p. 213]) is the matrix (ξi ) whose rows are indexed by
j
the irreducible characters and columns by the conjugacy classes. It takes the form ξi = χ j ( g i ). Now,
j
ξi = χ j ( g i ) = α ji .
Thus (13) gives the known formula for the structure constants for groups. That is
|C i ||C j | ξi ξ j ξt ∗
k k k
c ijt = .
|G | dk
k
For group algebras, it is obvious that the conjugacy classes Ci are stable under the adjoint action
of the group elements. We first generalize it for any Hopf algebra. We show:
Proposition 2.5. Let H be any finite-dimensional Hopf algebra, let z be a central element and I = z H ∗ the
left coideal generated by z. Then I is ad-stable.
˙ (z p ) =
had h1 s( p 1 ) z Sh2 s−1 ( p 3 ) p2
= h1 , s( p 1 ) h2 z s−1 ( p 3 ), Sh4 Sh3 p 2
= h 1 , s( p 1 ) p 3 , h 2 z p 2 (since z ∈ Z )
which is an element of I . 2
3230 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
Corollary 2.6. Let H be a semisimple Hopf algebra so that Coc( H ∗ ) is commutative. Then the conjugacy
classes Ci are stable under the adjoint action of H .
Another known property of finite group algebras (see e.g. [16]) is that the commutator subspace K
is given by
K= ag g a g = 0 for all conjugacy classes Ci .
g ∈G g ∈Ci
Theorem 2.7. Let H be a semisimple Hopf algebra so that Coc( H ∗ ) is commutative and let Ci be the conjugacy
class of H given as in (12). Then the commutator subspace K satisfies
K= (Ci ∩ ker ε ).
Proof. By Lemma 1.11(ii) K = (Coc( H ∗ ))0 . We show first that it contains Ci ∩ ker ε for all i. Observe
that
Ci ∩ ker ε = Λ F i p F i p , Λ = 0 . (14)
Since { F j } is a basis for Coc( H ∗ ) we need to show (Ci ∩ ker ε ), F j = 0 for all i, j. Indeed, since Λ
is cocommutative we have that if i = j, then Λ F i p , F j = 0. For i = j we have Λ F i p , F i =
Λ, F i p = 0 by assumption.
Conversely, let x ∈ (Coc( H ∗ ))0 . Since H ∗ = F i H ∗ it follows that H = Ci . Let x = xi , xi ∈ Ci ,
∗
we wish to show ε (xi ) = 0 for all i. Now, since x ∈ (Coc( H )) , we have i xi , F j = 0 for all j. Since
0
0= (Λ F i p i ), F j = Λ, F j p j = ε (x j ). 2
i
Lemma 2.8. Let H be any Hopf algebra (not necessarily finite-dimensional). Then K is a right H -module under
ah= Sh2 ah1 and a h = S −1 (h1 )ah2 .
Proof. We show that H is adl -stable when S 2 = Id. The other claims can be proved similarly. For all
h, x, y ∈ H ,
˙ (xy − yx) =
had h1 xy Sh2 − y ε (h)x − h1 yxSh2 − ε (h)xy − ε (h)(xy − yx) ∈ K . 2
˙ K = 0.
Corollary 2.9. Let H be a semisimple Hopf algebra. Then Λad
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3231
Theorems 2.1 and 2.3 are especially interesting when applied to semisimple
2 1 factorizable Hopf al-
gebras. Let ( H , R ) be a quasitriangular Hopf algebra. Set Q = R τ R = R r ⊗ R 1 r 2 . The maps f Q
and its dual f Q∗ are given by
f Q ( p) = p, R 2r1 R 1r2 , f Q∗ ( p ) = p, R 1r2 R 2r1 ,
si j = f Q (χi ), χ j .
Corollary 2.10. (See [8, Cor. 4.2].) The Verlinde formula for the fusion rules for factorizable semisimple Hopf
algebras:
coincides with the Verlinde formula for the fusion rules as given in Theorem 2.1(v).
Most importantly, we show in that in this case, the fusion rules and the structure constants are
equal up to rational scalars multiples.
Theorem 2.11. (See [8, Th. 4.3].) Let ( H , R ) is a factorizable semisimple Hopf algebra. Let χi , C j , di , αi j , cti j ,
mijt be as in Theorems 2.1 and 2.3. Then the structure constants c ijt are given by
di d j
c ijt = mijt .
dt
1
f Q (χi ) = Ci. (15)
di
Consider now the non-semisimple symmetric case. Observe first that since central elements act on
irreducible modules as scalars, we have χi Z = kχi for any irreducible character χi . Hence for any
symmetric algebra ( A , t ) with the corresponding A-module isomorphism φ : A → A ∗ , we have
φ φ −1 (χi ) Z = χi Z = kχi .
Lemma 3.1. Let ( A , t ) be a symmetric algebra. Then for any irreducible character χi , φ −1 (χi ) is nilpotent if and
only if V i is not projective. If V i is a projective irreducible module then (φ −1 (χi ))2 = α φ −1 (χi ), 0 = α ∈ k.
3232 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
Lemma 3.2. Let ( H , t ) is a symmetric Hopf algebra and let pa ∈ P be defined as in (2) and (3). Then:
pa , b = t τ (a), b = t , Λ1l−1 aS (Λ2 )b
= t , S (Λ2 )bΛ1l−1 a = t τ (b), a = p b , a.
(iii) Assume first that a satisfies ab = S 2 (b)a. Since l−1 a ∈ Z , it follows that for all b ∈ B,
pa , b = t , Λ1l−1 aS (Λ2 )b = t , l−1 aΛ1 S (Λ2 )b = 0.
The last equality follows since Λ1 S (Λ2 ) = ε (Λ) = 0.
If a satisfies ab = S −2 (b)a then S (a) satisfies S (a)b = S 2 (b) S (a) and the result follows from
part (ii). 2
Proof. Since u = S ( R 2 ) R 1 it follows that S (u ) = S(R1)S2(R2) = R 1 S ( R 2 ) hence u − S (u ) ∈ K .
−
Now, G = u v hence G
1 − 1
= uv − 1
and so G = S (G ) = S (u ) v . Thus G − G −1 = (u − S (u )) v −1 .
−1 −1
F := f Q Ψ,
F |2Z = S . (16)
Proposition 3.4. (See [7, Cor. 3.6].) Let ( H , R , v ) be a factorizable ribbon Hopf algebra and let E =
1
Λ ˙ F (G ). Then E ∈ Z 0 , that is, E H is a simple algebra.
dim( H ) ad
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3233
Factorizable ribbon Hopf algebras are symmetric since they are unimodular and S 2 is induced by
G −1 = uv −1 . Hence the central form t is given by
t = λ G −1 . (17)
φ S = sφ. (18)
f Q ( p ) = f Q ( p G ). (19)
This is done so f Q is an algebra isomorphism between Coc( H ∗ ) and Z in analogy with the
semisimple case. We “shift” the quantum Fourier transform as well.
(i)
f Q φ = F S and so
f Q φ = F −1 when restricted to Z .
(ii)
f Q s = S
fQ .
(iii) When restricted to Coc( H ∗ ) we have (φ
f Q )2 = s|Coc( H ∗ ) .
φ
f Q φ
f Q = φ(
f Q φ
by (18)
f Q φ)φ −1 = φ S φ −1 = s. 2
(ii) f Q (c G ) = f Q∗ (c G −1 ).
Proof. (i) By Lemma 1.11(ii), K = (Coc( H ∗ ))0 , which is a Z -submodule of H . The result follows now
from Lemma 3.3.
fQ =
(ii) By Proposition 3.5(ii), S f Q s when restricted to Coc( H ∗ ). Since f Q∗ = S f Q s and since
c ∈ Coc( H ∗ ) we have
f Q s(c ) =
f Q∗ c G −1 = S f Q s(c ) G = S f Q (c ). 2
c , G −1
f Q c = c G −1 , f Q c G = f Q∗ c G −1 , c G .
In analogy with the semisimple case (see (15)), the set { f Q (χ1 ), . . . ,
f Q (χn )} can be viewed as
a generalization of class sums and {f Q (χi ) H ∗ } as a generalization of conjugacy classes. They do
show similar properties as seen in the following.
Theorem 3.8. Let ( H , R , v ) be a factorizable ribbon Hopf algebra. For any irreducible character χi set
Ci = f Q (χi ) H ∗ .
Then:
where the right-hand side is easily seen to be a maximal ideal in H . Hence χi H is a simple
subcoalgebra for each i, and so their sum is direct; it is in fact the coradical of H ∗ . Since f Q is a
k-isomorphism, the result follows.
(iv)
Recall that the algebra of invariants of H under the adjoin action of H on itself is Z . Let
z= c i ∈ Z , where c i ∈ Ci for all i. Then had
˙ z = ε (h ) z = ε(h)c i for all h ∈ H . Since each Ci is
ad-stable by (i) and since their sum is direct, it follows that had ˙ c i = ε (h )c i , hence c i ∈ Z for all i.
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3235
To see the last part, observe that since χi H is a simple coalgebra, it follows that its space of
cocommutative elements is 1-dimensional, and thus equals kχi . Since only Coc( H ∗ ) is mapped un-
der
f Q to Z , it follows that
f Q (χi H ) ∩ Z is 1-dimensional as well, implying the desired result. 2
Another property of conjugacy classes in the semisimple case is their connection to K , the com-
mutator subspace of H . We can show the following:
Proof. By Theorem 3.8(ii) we may assume that G f Q (χi h) ∈ ker ε , for some h ∈ H , that is,
χi , Gh = 0. By Lemma 1.11(ii), we need to show that G f Q (χi h), c = 0 for all c ∈ Coc( H ∗ ).
Now, by Lemma 3.7(i), this is equivalent to χi h, G f Q (c ) = 0. Let { E j } be the set of all primitive
idempotents of Z and note that {
f Q−1 ( E j )} is the set of all primitive idempotents of Coc( H ∗ ). Only
one E satisfies χi E = χi , and for c = f Q−1 ( E ) we have
χi h, Gf Q (c ) = χi , hG E = χi , hG = 0
by assumption on h. If c =
f Q−1 ( E ) where E satisfies χi E = 0, then
χi h, Gf Q (c ) = χi h, G E = 0.
If c ∈ J (Coc( H ∗ )) then G
f Q (c ) ∈ J ( Z ) ⊂ J ( H ), hence
χi h, Gf Q (c ) = χi , hGf Q (c ) = 0.
Since Coc( H ∗ ) is spanned over k by the set of all primitive central idempotents + J (Coc( H ∗ )) we
are done. 2
Crucial ingredients in the theory of invariants of knots and 3-manifolds are quantum dimensions
and quantum traces. In what follows we discuss these notions for the character ring R, reminiscent
of the methods used to derive modular Hopf algebras.
Recall that the categorical definitions of quantum dimension of a module V and the quantum trace
of a map T in a ribbon category [2], translates to the following:
qtrace( T ) = Trace ρ V (G ) T , qdim( V ) = qtrace(Id V ) = χ V , G .
We say that the module V has quantum trace 0 if the quantum trace of T equals 0 for all A-
module maps T .
The first part of the following is well known, the second one appears in the literature without
explicitly stating it as a result (it essentially follows from arguments in the proof of [18, Th. 2.3(b)]).
We give a proof based on the setup of the previous sections.
qdim( V ) = 0 ⇔ qtrace( V ) = 0 ⇔ χ V , G Z = 0.
(ii) If H is not semisimple then any projective H -module V satisfies qtrace( V ) = 0. In particular qdim( V ) =
χ V , G = 0 and for all c ∈ P , c , G Z = 0.
3236 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
Proof. (i) The first equivalence follows since any A-module map on V is a scalar multiplication of Id.
The second one follows since elements of Z act on V as scalars.
(ii) Any indecomposable projective module has the form He where e is a primitive idempotent
of H . Any A-module map T on He is given by right multiplication by eae, a ∈ H . By Remark 1.3,
χ Ae = p e . Hence
Trace ρ Ae (G ) T = Trace l(G ) ◦ r (eae) = p eae , G = 0.
In what follows we lay the foundations for the distinction between “well-behaved” and “poorly-
behaved” irreducible characters according the vanishing of their quantum dimension. We add to the
chain given in (8), a new component T containing the ideal P of characters of projective modules.
Set
T = c ∈ R c , G
f Q (R) = 0 . (21)
The last equality follows from (20) and since R is s-stable. This implies by definition of T that
φf Q (R) ∩ R ⊂ T .
(iv) Since
f Q is an algebra map we have
f Q ( J (R)) ⊂ J ( Z ) ⊂ J ( H ). Hence
J (R), G
f Q (R) = G
f Q J (R) , R = 0. 2
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3237
M j = k χk
f Q ( F j ) = χk . (22)
0 = G
f Q ( F j ), R = F j , G
f Q (R) .
Hence F j ∈ T . 2
(i) If l, l ∈ M j and χl , χl ∈
/ T , then
qd l−1 χl − qd l− 1 χl ∈ T .
(ii) For each M j choose χl j ∈ M j so that χl j ∈/ T , then the set {χl j , 1 j m} is linearly independent
modulo T .
s
f Q (R) = k
f Q (F j) ⊕ N , N ⊂ J ( H ).
j =1
qd l− 1 χl − qd l−1 χl , G
f Q ( F j ) = 0.
m
(ii) If j =1 αl j χl j ∈ T , then for each j,
m
0= αl j χl j , Gf Q ( F j ) = αl j qd l j ,
j =1
implying αl j = 0. 2
Renumber the χ ’s so that χ j satisfies j ∈ M j , and χ j ∈/ T . Proposition 3.13 implies now that R/T
has two bases that are the corresponding images of
B = { F 1 , . . . , F m } and C = {χ1 , . . . , χm }.
By Lemma 3.7(i),
G
f Q (χi ), F j = χi , Gf Q ( F j ) = δi j qd i ,
m
χi = χi , Gf Q (χ j ) qd −j 1 F j + T . (23)
j =1
Set
s ji = χi , Gf Q (χ j ) qd −j 1 , (u i j ) = (si j )−1 (24)
and
s̃ ji = χi , Gf Q (χ j ) . (25)
Then we have:
Theorem 3.14. Let ( H , R , v ) be a factorizable ribbon Hopf algebra over an algebraically closed field of char-
acteristic zero. Let T be defined as in (21) and M j as in (22). Let { F 1 , . . . , F m } be the set of all primitive
idempotents of R so that F j ∈ / T for all j, and χ j so that j ∈ M j and χ j ∈
/ T . Let the matrix (si j ) be defined
as in (24) and (s̃i j ) as in (25). Then both matrices are invertible and the matrix (s̃i j ) is symmetric.
Proof. The matrix (s̃i j ) is symmetric by Lemma 3.7(i). Both matrices are invertible since (si j ) is the
change of bases matrix given in (23), and (s̃i j ) is obtained by multiplying (si j ) from the right by
invertible diagonal matrices. 2
Recall, for any Hopf algebra the product χ V χW is the character χ V ⊗W , hence by Proposition 1.1(iv),
for all 1 i , l n we have
n
i
χi χl = mkl χk . (26)
k =1
M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240 3239
For 1 j m, 1 i , l n, set
m̃ijl = i
mkl qd k . (27)
k∈M j
Note that since G k = 1 for some integer k, it follows that for any module V , ρ V (G ) is diagonaliz-
j
able with eigenvalues roots of unity. Hence χ V , G is an algebraic integer and so are all m̃il .
Consider now that map l(χi ) : R/T → R/T , defined as multiplication by χi . By Proposition 3.13(i)
we modify (26) as follows:
m
χi χl = m̃ijl qd −
j
1
χj + T . (28)
j =1
Thus the matrix of l(χi ) with respect to the basis C of R/T is Mi where
Mi jl
= qd −j 1m̃ijl , (29)
1 j , l m. Observe that by (23), F j are eigenvectors for l(χi ) with corresponding eigenvalues s ji .
Moreover, by (23) we have
m
m
χi χl = sti stl F t + T = sti stl u jt χ j + T .
t =1 t , j =1
Hence:
m
m̃ijl = qd j sti stl u jt . (30)
t =1
We summarize:
Theorem 3.15. Let ( H , R , v ) be a factorizable ribbon Hopf algebra over an algebraically closed field of char-
acteristic zero. Let T be defined as in (21) and M j as in (22). Let { F 1 , . . . , F m } be the set of all primitive
idempotents of R so that F j ∈/ T and χ j so that j ∈ M j and χ j ∈/ T . Let A = (si j ), A −1 = (u i j ) be the change
of bases matrices given in (24). Let m̃ jl be given in (27), M in (29) and let Di = diag{si1 , si2 , . . . , sim }. Then
i i
for each 1 i m:
D i = A −1 M i A .
m
m̃ijl = qd j sti stl u jt .
t =1
3240 M. Cohen, S. Westreich / Journal of Algebra 324 (2010) 3219–3240
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