Definitions

Derivations on an algebra

For $A$ an algebra (over some ring $k$), a derivation on $A$ is a $k$-linear morphism

such that for all $a,b \in A$ we have

This identity is called the Leibniz rule; compare it to the product rule in ordinary calculus (first written down by Gottfried Leibniz).

In a cancellative algebra

For $A$ a cancellative algebra, where for all $c \neq 0$, $c a = c b$ implies $a = b$ and $a c = b c$ implies $a = b$, the derivation satisfies the following identity: for any scalar $c$ in $k$, $d(c) = 0$. This follows from $k$-linearity, which says that for any scalar $c$ in $k$ and for any non-zero element $f$ in $A$, $d(c f) = c d(f)$ and $d(f c) = d(f) c$. The Leibniz rule states that $d(c f) = c d(f) + d(c) f$ and $d(f c) = d(f) c + f d(c)$. It follows that $d(c) f = 0$ and $f d(c) = 0$, and due to the cancellative property and the fact that $f$ is non-zero, it follows that $d(c) = 0$.

In a division algebra

For $A$ a division algebra, with left division $\backslash$ and right division $/$ the derivation satisfies the left and right quotient identities for $f$ in $A$ and non-zero $g$ in $A$:

and

which reduces to the regular quotient rule? if the division algebra is associative and commutative:

Derivations with values in a bimodule

For $A$ an algebra (over some ring $k$) and $N$ a bimodule over $A$, a derivation of $A$ with values in $N$ is a $k$-linear morphism

such that for all $a,b \in A$ we have

where on the dot on the right-hand side denotes the right (first term) and left (second term) action of $A$ on the bimodule $N$.

The previous definition is a special case of this one, where the bimodule is $N = A$, the algebra itself with its canonical left and right action on itself.

A special case is where $A$ is a group algebra $k G$ of a group $G$, and $N$ is a left $G$-module, regarded as a bimodule where the right action is trivial. Here a derivation is also called a 1-cocycle, as used in group cohomology.

Graded derivations

A graded derivation of degree $p$ on a graded algebra $A$ is a degree-$p$ graded-module homomorphism $d: A \to A$ such that

whenever $a$ is homogeneous of degree $q$. (By default, the grade is usually $1$, or sometimes $-1$.)

Augmented derivations

An augmented derivation on an algebra $A$, augmented by an algebra homomorphism $\epsilon: A \to B$, is a module homomorphism $d: A \to B$ such that

If you think about it, you should be able to figure out the definition of an augmented graded derivation.

Further variations

There are many further extensions, for examples derivations with values in an $A$-bimodule $M$ forming $Der_k(A,M) \subset Hom_k(A,M)$ (see also double derivation), skew-derivations in ring theory (with a twist in the Leibniz rule given by an endomorphism of a ring) and the dual notion of a coderivation of a coalgebra. The latter plays role in Koszul-dual definitions of $A_\infty$-algebras and $L_\infty$-algebras. See also derivation on a group, which uses a modified Leibniz rule: $d(a b) = d(a) + a d(b)$.

Derivations on algebras over a dg-operad

More generally, there is a notion of derivation for every kind of algebra over an operad over a dg-operad (at least).

This appears as (Hinich, def. 7.2.1).

The theory of tangent complexes, Kähler differentials, etc. exists in this generality for derivations on algebras over an operad.

Generalization to arbitrary $(\infty,1)$-categories

Another equivalent reformulation of the notion of derivations turns out to be useful for the vertical categorification of the concept:

for $N$ an $R$-module, there is the nilpotent extension ring $G(N) := N \oplus R$, equipped with the product operation

This comes with a natural morphism of rings

given by sending the elements of $N$ to 0. One sees that a derivation on $R$ with values in $N$ is precisely a ring homomorphism $R \to G(N)$ that is a section of this morphism.

In terms of the bifibration $p : Mod \to Ring$ of modules over rings, this is the same as a morphism from the module of Kähler differentials $\Omega_K(R)$ to $N$ in the fiber of $p$ over $R$.

While this is a trivial restatement of the universal property of Kähler differentials, it is this perspective that vastly generalizes:

we may replace $Mod \to Ring$ by the tangent (∞,1)-category projection $p : T_C \to C$ of any (∞,1)-category $C$. The functor that assigns Kähler differentials is then replaced by a left adjoint section of this projection

An $(\infty,1)$-derivation on an object $R$ with coefficients in an object $N$ in the fiber of $T_C$ over $R$ is then defined to be morphism $\Omega(R) \to N$ in that fiber.

More discussion of this is at deformation theory.

Examples

Derivations on an algebra

• Let $A$ consist of the smooth real-valued functions on an interval in the real line. Then differentiation is a derivation; this is the motivating example.
• Let $A$ consist of the holomorphic functions on a region in the complex plane. Then differentiation is a derivation again.
• Let $A$ consist of the meromorphic functions on a region in the complex plane. Then differentiation is still a derivation.
• Let $A$ consist of the smooth functions on a manifold (or generalized smooth space) $X$. Then any tangent vector field on $X$ defines a derivation on $A$; indeed, this serves as one definition of tangent vector field.
• Let $A$ consist of the germs of differentiable functions near a point $p$ in a smooth space $X$. Then any tangent vector at $a$ on $X$ defines a derivation on $A$ augmented by evaluation at $a$; again, this serves to define tangent vectors.
• Let $A$ consist of the smooth differential forms on a smooth space $X$. Then exterior differentiation is a graded derivation (of degree $1$).
• In any of the above examples containing the adjective ‘smooth’, replace it with $C^k$ and augment $A$ by the inclusion of $C^k$ into $C^{k-1}$. Then we have an augmented derivation.

More algebraic examples:

• Let $A$ be any algebra over a ring. The constant function $D(a) = 0$ for all $a \in A$ is a derivation.

• Let $R[[x]]$ be a formal power series over a ring $R$. Then the formal derivative? is a derivation.

• A less obvious one: Let $R[X_{1},...,X_{n}]$ be a polynomial algebra over a commutative rig $R$. Define $L(P)=\underset{1 \le k \le n}{\sum}\frac{\partial P}{\partial X_{i}}X_{i}$. Then $L:R[X_{1},...,X_{n}] \rightarrow R[X_{1},...,X_{n}]$ is a derivation. Moreover, if we suppose that $R$ is a multiplicatively cancellable rig, then the homogeneous polynomials of degree $n$ are exactly the solutions of the differential equation $L(P)=n.P$ (see the page homogeneous polynomials).

Derivations with values in a bimodule

The standard example of a derivation not on an algebra, but with values in a bimodule is a restriction of the above case of the exterior differential acting on the deRham algebra of differential forms. Restricting this to 0-forms yields a morphism

where $\Omega^1(X)$ is the space of 1-forms on $X$, regarded as a bimodule over the algebra of functions in the obvious way.

A variation of this example is given by the Kähler differentials. These provide a universal derivation in some sense.

Derivations of smooth functions

See also at derivations of smooth functions are vector fields.

Derivations of continuous functions

Let now $X$ be a topological manifold and $C(X)$ the algebra of continuous real-valued functions on $X$.

Related concepts

• operational tangent space

• p-derivation

• derivation Lie algebra

• sequence algebra

• Hasse-Schmidt derivation

References

Derivations on algebras over a dg-operad are discussed in section 7 of

• Vladimir Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10). 3291-3323 (1997)(arXiv:q-alg/9702015, Erratum (arXiv:math/0309453))