graded derivation (changes)

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If we have a graded algebra AA, and DD is a homogeneous linear map of grade d=|deg(D | ) 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^{|a||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 sameϵ\epsilon.

If the commutator factorϵ=1\epsilon = 1 , this definition reduces to the usual Leibniz case. rule. Ifϵ=1\epsilon = -1, however, then D(ab)=D(a)b+(1) |a|aD(b)D(ab)=D(a)b+(-1)^{|a|}aD(b), for odd |D||D|. They are called anti-derivations.

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.