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 ${\displaystyle A}$ over a ring or a field ${\displaystyle K}$, a ${\displaystyle K}$-derivation is a ${\displaystyle K}$-linear map ${\displaystyle D:A\to A}$ that satisfies Leibniz's law:

${\displaystyle D(ab)=aD(b)+D(a)b.}$

More generally, if ${\displaystyle M}$ is an ${\displaystyle A}$-bimodule, a ${\displaystyle K}$-linear map ${\displaystyle D:A\to M}$ that satisfies the Leibniz law is also called a derivation. The collection of all ${\displaystyle K}$-derivations of ${\displaystyle A}$ to itself is denoted by ${\displaystyle \mathrm {Der} _{K}(A)}$. The collection of ${\displaystyle K}$-derivations of ${\displaystyle A}$ into an ${\displaystyle A}$-module ${\displaystyle M}$ is denoted by ${\displaystyle \mathrm {Der} _{K}(A,M)}$.

Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an ${\displaystyle \mathbb {R} }$-derivation on the algebra of real-valued differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$. The Lie derivative with respect to a vector field is an ${\displaystyle \mathbb {R} }$-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 ${\displaystyle A}$ is noncommutative, then the commutator with respect to an element of the algebra ${\displaystyle A}$ defines a linear endomorphism of ${\displaystyle A}$ to itself, which is a derivation over ${\displaystyle K}$. That is,

${\displaystyle [FG,N]=[F,N]G+F[G,N]\,,}$

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