graded derivation

If we have a graded algebra AA, and DD is a homogeneous linear map of grade d=deg(D)d = deg(D) on AA then DD is a homogeneous derivation if

D(ab)=D(a)b+ϵ deg(a)deg(D)aD(b)D(ab)=D(a)b+\epsilon^{deg(a)\cdot deg(D)}aD(b)

ϵ{1,1}\epsilon\in\{-1,1\} acting on homogeneous elements of AA. A graded derivation is a sum of homogeneous derivations with the same ϵ\epsilon.

If ϵ=1\epsilon = 1, this definition reduces to the Leibniz rule.

If ϵ=1\epsilon = -1, then D(ab)=D(a)b+(1) |a|aD(b)D(ab)=D(a)b+(-1)^{|a|}aD(b), for odd |D||D|. The notion of graded derivations of odd degree is sometimes called antiderivation or anti-derivation or integration.

Examples of anti-derivations include the exterior derivative? and the interior product? acting on differential form?s.

Last revised on August 18, 2012 at 20:27:43. See the history of this page for a list of all contributions to it.