#
nLab
graded commutator

### Context

#### Super-Algebra and Super-Geometry

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 17:04:56.
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