Derivation (abstract Algebra) - Graded Derivations

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. Z₂-graded algebras) are often called superderivations.