nLab
graded commutator

In superalgebra the the graded commutator or supercommutator of two elements of homogeneous degree is

[a,b](ab(1) deg(a)deg(b)ba) [a,b] \coloneqq \left( a b - (-1)^{deg(a) deg(b)} b a\right)

and extended from there to the whole superalgebra as a graded derivation? in both arguments.

So when at least one of aa or bb is even graded, then this is the commutator. When both are odd graded then this is the anti-commutator.

The graded commutator is just the plain commutator internal to the symmetric monoidal category of super vector spaces.

Created on November 24, 2014 at 17:09:45. See the history of this page for a list of all contributions to it.