Supercommutator - Knowlpedia

Core idea

Let $A$ be an associative superalgebra. For homogeneous $a,b\in A$ , their supercommutator is

[a,b]_{\mathrm s} =ab-(-1)^{|a||b|}ba.

It extends bilinearly to all of $A$ . This bracket is graded-skew and satisfies the super Jacobi identity, so it turns $A$ into a Lie superalgebra.

If either element is even, this is the usual commutator. If both are odd, then

[a,b]_{\mathrm s}=ab+ba,

so their supercommutator is their ordinary anticommutator.

References

M. Scheunert, The Theory of Lie Superalgebras, Lecture Notes in Mathematics 716, Springer, 1979. Publisher record. Relevant: Chapter 1.
I. M. Musson, Lie Superalgebras and Enveloping Algebras, American Mathematical Society, 2012. Publisher record. Relevant: Chapter 1.