graded commutator

**superalgebra** and (synthetic ) **supergeometry**

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

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

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

So when at least one of $a$ or $b$ 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.