Derivation (abstract algebra)

In abstract algebra, a derivation on an algebra A over a ring or 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.

Antiderivation

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.

See also

• Differential Galois theory
• Kähler differential