nLab graded derivation






The term graded derivation would a priori refer in general to suitably compatible derivations of graded algebras, but is in practice used mostly for those operations d:AAd \,\colon\, A \to A on superalgebras AA that satisfy the condition that for a 1,a 2Aa_1, a_2 \,\in\, A a pair of elements each of homogeneous degree deg(a i)mod2deg(a_i) \in \mathbb{Z} \,mod\, 2 the derivation of their product is

d(a 1a 2)=(da 1)a 2+(1) deg(a 1)deg(d)a 1da 2, d(a_1 \cdot a_2) \;=\; (d a_1) \cdot a_2 + (-1)^{deg(a_1) \cdot deg(d)} a_1 \cdot d a_2 \,,

where deg(d)mod2deg(d) \,\in\, \mathbb{Z} \,mod\,2 is the degree of the graded derivation.

For d=0d = 0 this is the law satisfied by an ordinary (un-graded) derivation, also known as the Leibniz rule satisfied by ordinary differentiation. The archetypical example of a non-trivially graded derivation is the de Rham differential acting on the de Rham algebra.

Created on May 3, 2023 at 05:10:10. See the history of this page for a list of all contributions to it.