Derivation (abstract algebra)
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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).
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.
[edit] Graded derivations
If we have a graded algebra A, and D is a grade |D| linear map on A then D is a graded derivation if
- D(ab) = D(a)b + ( − 1) | a | | D | aD(b)
acting on homogeneous elements of A. If |D| is even this definition reduces to the usual case. If D is odd, however, it obeys the rule:
- D(ab) = (Da)b + (−1)|a|a(Db).
Odd derivations are often called antiderivations.
Examples of odd derivations include the exterior derivative and the interior product acting on differential forms.
Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations.
[edit] See also
- In elemental differential geometry derivations are tangent vectors
- Differential Galois theory
- Kähler differential
