# Derivation (differential algebra)

# Derivation (differential algebra)

In mathematics, a **derivation** is a function on an algebra which 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:

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 Der*K*(*A*). The collection of *K*-derivations of *A* into an *A*-module *M* is denoted by 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 **R**-derivation on the algebra of real-valued differentiable functions on **R***n*. 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*. 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, andis a*K*-derivation, thenIf

*A*has a unit 1, then*D*(1) =*D*(12) = 2*D*(1), so that*D*(1) = 0. Thus by*K*-linearity,*D*(*k*) = 0 for allIf

*A*is commutative,*D*(*x*2) =*xD*(*x*) +*D*(*x*)*x*= 2*xD*(*x*), and*D*(*x**n*) =*nx**n*−1*D*(*x*), by the Leibniz rule.More generally, for any

*x*1,*x*2, ...,*x**n*∈*A*, it follows by induction that

*D**n*is not a derivation, instead satisfying a higher-order Leibniz rule:

- Moreover, if

*M*is an

*A*-bimodule, write

*K*-derivations from

*A*to

*M*.

Der

*K*(*A*,*M*) is a module over*K*.Der

*K*(*A*) is a Lie algebra with Lie bracket defined by the commutator:

There is an

*A*-module (called the Kähler differentials) with a*K*-derivation through which any derivation factors. That is, for any derivation*D*there is a*A*-module map with

*A*-modules:

If

*k*⊂*K*is a subring, then*A*inherits a*k*-algebra structure, so there is an inclusion

*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

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* = −1, however, then

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. **Z**2-graded algebras) are often called **superderivations**.

Related notions

Hasse–Schmidt derivations are *K*-algebra homomorphisms

`Composing further with the map which sends aformal power seriesto the coefficientgives a derivation.`

See also

In differential geometry derivations are tangent vectors

Kähler differential

Hasse derivative

p-derivation

Wirtinger derivatives

Derivative of the exponential map