# Derivation (abstract algebra)

In Differential algebra, an area of mathematics devoted to the algebraic study of differential equations, 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*-module, a *K*-linear map *D*:*A*→*M* which 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. 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

The Leibniz law itself has a number of immediate consequences. Firstly, if *x*_{1}, *x*_{2}, … ,*x*_{n} ∈ *A*, then it follows by mathematical induction that

In particular, if *A* is commutative and *x*_{1} = *x*_{2} = … = *x*_{n}, then this formula simplifies to the familiar power rule *D*(*x*^{n) = nxn−1D(x). Secondly, if A has a unit element 1, then D(1) = 0 since D(1) = D(1·1) = D(1) + D(1). Moreover, because D is K-linear, it follows that “the derivative of any constant function is zero”; more precisely, for any x ∈ K, D(x) = D(x·1) = x·D(1) = 0.
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If *k* ⊂ *K* is a subring, and *A* is a *k*-algebra, then there is an inclusion

since any *K*-derivation is *a fortiori* a *k*-derivation.

The set of *k*-derivations from *A* to *M*, Der_{k}(*A*,*M*) is a module over *k*. Furthermore, the *k*-module Der_{k}(*A*) forms a Lie algebra with Lie bracket defined by the commutator:

It is readily verified that the Lie bracket of two derivations is again a derivation.

## Graded derivations

If we have a graded algebra *A*, and *D* is a homogeneous linear map of grade *d* = |*D*| on *A* then *D* is a **homogeneous derivation** if
, ε = ±1
acting on homogeneous elements of *A*. A **graded derivation** is sum of homogeneous derivations with the same ε.

If the commutator factor ε = 1, this definition reduces to the usual case. If ε = −1, however, then , for odd |*D*|. They are called **anti-derivations**.

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**.

## See also

- In elemental differential geometry derivations are tangent vectors
- Kähler differential
- Hasse derivative
- p-derivation
- Wirtinger derivatives

## References

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