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

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

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

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

If $\epsilon = -1$, then $D(ab)=D(a)b+(-1)^{|a|}aD(b)$, for odd $|D|$. 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.