# 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, and is a *K*-derivation, then

- If
*A*has a unit 1, then*D*(1) =*D*(1^{2}) = 2*D*(1), so that*D*(1) = 0. Thus by*K*-linearity,*D*(*k*) = 0 for all - If
*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

- which is if for all commutes with .

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

- Moreover, if
*M*is an*A*-bimodule, write - for the set of
*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:

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

- 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

- The correspondence is an isomorphism of
*A*-modules:

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

- 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

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 a formal power series to the coefficient gives a derivation.

## See also

## References

- Bourbaki, Nicolas (1989),
*Algebra I*, Elements of mathematics, Springer-Verlag, ISBN 3-540-64243-9. - Eisenbud, David (1999),
*Commutative algebra with a view toward algebraic geometry*(3rd. ed.), Springer-Verlag, ISBN 978-0-387-94269-8. - Matsumura, Hideyuki (1970),
*Commutative algebra*, Mathematics lecture note series, W. A. Benjamin, ISBN 978-0-8053-7025-6. - Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993),
*Natural operations in differential geometry*, Springer-Verlag.