Derivation (abstract algebra)

In abstract algebra, a derivation on an algebra A over a field k is a linear map

D : A → A

that satisfies Leibniz' law:

D(ab) = (Da)b + a(Db).

As a consequence, if A is unital,

then

D(1) = 0 since

D1 = D(1·1) = D1 + D1.

Examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme.

See also: Kähler differential

If we have a Z₂ graded algebra A, D is an antiderivation if

D(ab) = (Da)b + (−1)^deg(a)a(Db).

The same proof showing D(1)=0 applies, if A is unital.