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.

If ${\displaystyle A}$ is a ${\displaystyle K}$-algebra, for ${\displaystyle K}$ a ring, and D: ${\displaystyle A}$ → ${\displaystyle A}$ is a ${\displaystyle K}$-derivation, then

• If ${\displaystyle A}$ has a unit 1, then ${\displaystyle D(1)=D(1^{2})=2D(1)}$, so that ${\displaystyle D(1)=0}$. Thus by ${\displaystyle K}$-linearity, ${\displaystyle D(k)=0}$ for all ${\displaystyle k\in K}$.
• If ${\displaystyle A}$ is commutative, then ${\displaystyle D(x^{2})=xD(x)+D(x)x=2xD(x)}$, and ${\displaystyle D(x^{n})=nx^{n-1}D(x)}$, by the Leibniz rule.
• More generally, for any ${\displaystyle x_{1},x_{2},\ldots ,x_{n}\in A}$, it follows by induction that

  ${\displaystyle D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}}$

which is ${\displaystyle \textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}}$ if for all ${\displaystyle i}$, ${\displaystyle D(x_{i})}$ commutes with ${\displaystyle x_{1},x_{2},\ldots ,x_{i-1}}$.

• For ${\displaystyle n>1}$, ${\displaystyle D^{n}}$ is not a derivation, instead satisfying a higher-order Leibniz rule:

${\displaystyle D^{n}(uv)=\sum _{k=0}^{n}{\binom {n}{k}}\cdot D^{n-k}(u)\cdot D^{k}(v).}$

Moreover, if ${\displaystyle M}$ is an ${\displaystyle A}$-bimodule, write

${\displaystyle \operatorname {Der} _{K}(A,M)}$

for the set of ${\displaystyle K}$-derivations from ${\displaystyle A}$ to ${\displaystyle M}$.

• ${\displaystyle \mathrm {Der} _{K}(A,M)}$ is a module over ${\displaystyle K}$.
• ${\displaystyle \mathrm {Der} _{K}(A)}$ is a Lie algebra with Lie bracket defined by the commutator:

${\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.}$

since it is readily verified that the commutator of two derivations is again a derivation.

• There is an ${\displaystyle A}$-module ${\displaystyle \Omega _{A/K}}$ (called the Kähler differentials) with a ${\displaystyle K}$-derivation ${\displaystyle d:A\to \Omega _{A/K}}$ through which any derivation ${\displaystyle D:A\to M}$ factors. That is, for any derivation ${\displaystyle D}$' there is a ${\displaystyle A}$-module map ${\displaystyle \varphi }$ with

${\displaystyle D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M}$

The correspondence ${\displaystyle D\leftrightarrow \varphi }$ is an isomorphism of ${\displaystyle A}$-modules:

${\displaystyle \operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)}$

• If ${\displaystyle k\subset K}$ is a subring, then ${\displaystyle A}$ inherits a ${\displaystyle k}$-algebra structure, so there is an inclusion

${\displaystyle \operatorname {Der} _{K}(A,M)\subset \operatorname {Der} _{k}(A,M),}$

since any ${\displaystyle K}$-derivation is a fortiori a ${\displaystyle k}$-derivation.

Given a graded algebra ${\displaystyle A}$ and a homogeneous linear map ${\displaystyle D}$ of grade ${\displaystyle |D|}$ on ${\displaystyle A}$, ${\displaystyle D}$ is a homogeneous derivation if

${\displaystyle {D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b)}}$

for every homogeneous element ${\displaystyle A}$ and every element ${\displaystyle b}$ of ${\displaystyle A}$ for a commutator factor ${\displaystyle \varepsilon =\pm 1}$. A graded derivation is sum of homogeneous derivations with the same ${\displaystyle \varepsilon }$.

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

${\displaystyle {D(ab)=D(a)b+(-1)^{|a||D|}aD(b)}}$

for odd ${\displaystyle |D|}$, and ${\displaystyle D}$ 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., ${\displaystyle \mathbb {Z} _{2}}$-graded algebras) are often called superderivations.

Hasse–Schmidt derivations are ${\displaystyle K}$-algebra homomorphisms

${\displaystyle A\to A[[t]].}$

Composing further with the map that sends a formal power series ${\displaystyle \sum a_{n}t^{n}}$ to the coefficient ${\displaystyle a_{1}}$ gives a derivation.