Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law:


 * $$ D(ab) = a D(b) + D(a) b.$$

More generally, if M is an A-bimodule, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M).

Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. That is,
 * $$[FG,N]=[F,N]G+F[G,N],$$

where $$[\cdot,N]$$ is the commutator with respect to $$N$$. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

Properties
If A is a K-algebra, for K a ring, and $D: A → A$ is a K-derivation, then
 * If A has a unit 1, then D(1) = D(12) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all $k ∈ K$.
 * If A is commutative, D(x2) = xD(x) + D(x)x = 2xD(x), and D(xn) = nxn−1D(x), by the Leibniz rule.
 * More generally, for any $x_{1}, x_{2}, …, x_{n} ∈ A$, it follows by induction that
 * $$D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n $$
 * which is $\sum_i D(x_i)\prod_{j\neq i}x_j$ if for all $i$, $D(x_{i})$ commutes with $$x_1,x_2,\ldots, x_{i-1}$$.


 * For n > 1, Dn is not a derivation, instead satisfying a higher-order Leibniz rule:
 * $$D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot D^k(v).$$
 * Moreover, if M is an A-bimodule, write
 * $$ \operatorname{Der}_K(A,M)$$
 * for the set of K-derivations from A to M.


 * DerK(A, M) is a module over K.
 * DerK(A) is a Lie algebra with Lie bracket defined by the commutator:
 * $$[D_1,D_2] = D_1\circ D_2 - D_2\circ D_1.$$
 * since it is readily verified that the commutator of two derivations is again a derivation.


 * There is an A-module $Ω_{A/K}$ (called the Kähler differentials) with a K-derivation $d: A → Ω_{A/K}$ through which any derivation $D: A → M$ factors. That is, for any derivation D there is a A-module map $φ$ with
 * $$ D: A\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M $$
 * The correspondence $$ D\leftrightarrow \varphi$$ is an isomorphism of A-modules:
 * $$ \operatorname{Der}_K(A,M)\simeq \operatorname{Hom}_{A}(\Omega_{A/K},M)$$


 * If $k ⊂ K$ is a subring, then A inherits a k-algebra structure, so there is an inclusion
 * $$\operatorname{Der}_K(A,M)\subset \operatorname{Der}_k(A,M) ,$$
 * since any K-derivation is a fortiori a k-derivation.

Graded derivations
Given a graded algebra A and a homogeneous linear map D of grade $|D|$ on A, D is a homogeneous derivation if
 * $${D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)}$$

for every homogeneous element a and every element b of A for a commutator factor ε = ±1. A graded derivation is sum of homogeneous derivations with the same ε.

If ε = 1, this definition reduces to the usual case. If ε = &minus;1, however, then
 * $${D(ab)=D(a)b+(-1)^{|a|}aD(b)}$$

for odd $|D|$, and D is called an anti-derivation.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.

Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations.

Related notions
Hasse–Schmidt derivations are K-algebra homomorphisms


 * $$A \to At.$$

Composing further with the map which sends a formal power series $$\sum a_n t^n$$ to the coefficient $$a_1$$ gives a derivation.