Spahn
graded derivation (Rev #1, changes)

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

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

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

If the commutator factor ϵ=1\epsilon = 1, this definition reduces to the usual case. If ϵ=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.

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

Revision on August 18, 2012 at 16:07:57 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.