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

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

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

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

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.