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Derivation (differential algebra)

Algebraic generalization of the derivative

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated April 2, 2026

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\to A that satisfies Leibniz's law:

D(ab)=aD(b)+D(a)b.

More generally, if M is an A-bimodule, a K-linear map D:A\to M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by \mathrm {Der} _{K}(A). The collection of K-derivations of A into an A-module M is denoted by \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 \mathbb {R}-derivation on the algebra of real-valued differentiable functions on \mathbb {R} ^{n}. The Lie derivative with respect to a vector field is an \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 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.

01Properties

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(1^{2})=2D(1), so that D(1)=0. Thus by K-linearity, D(k)=0 for all k\in K.
  • If A is commutative, then D(x^{2})=xD(x)+D(x)x=2xD(x), and D(x^{n})=nx^{n-1}D(x), by the Leibniz rule.
  • More generally, for any x_{1},x_{2},\ldots ,x_{n}\in A, it follows by induction that
    D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}
which is \textstyle \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, D^{n} 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.
[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 \Omega _{A/K} (called the Kähler differentials) with a K-derivation d:A\to \Omega _{A/K} through which any derivation D:A\to M factors. That is, for any derivation D' there is a A-module map \varphi 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\subset 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.

02Graded 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 \varepsilon =\pm 1. A graded derivation is sum of homogeneous derivations with the same \varepsilon.

If \varepsilon =1, this definition reduces to the usual case. If \varepsilon =-1, however, then

{D(ab)=D(a)b+(-1)^{|a||D|}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., \mathbb {Z} _{2}-graded algebras) are often called superderivations.

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Sources and credits

This article is adapted from the Wikipedia article “Derivation (differential algebra)”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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