---
tag: 6JZW4JC68Z
kind: concept
name: Superalgebra
msc: [17A70, 16W55, 16W50]
sources:
  - url: https://en.wikipedia.org/wiki/Superalgebra
same_as:
  wikidata: Q3977654
  wikipedia: Superalgebra
---

## Summary

A superalgebra is an algebra graded by $\Z/2\Z$: it splits as an even part and an odd part, and the product adds parities.
It is the algebraic setting for supersymmetry, where even elements behave like ordinary commuting quantities and odd elements like anticommuting ones.
The [[exterior algebra|0MDMZ965VE]] and [[Clifford algebras|A38B1QFK20]] are basic examples, and Lie superalgebras are the corresponding nonassociative notion.
Many constructions, such as tensor products and commutators, acquire a sign $(-1)^{\abs{a}\abs{b}}$ when two odd elements are moved past each other.

## Explanations

### L1 human

Sort the elements of an algebra into two kinds, even and odd, like even and odd numbers.
The product follows the rule for adding parities: even times even and odd times odd are even, and even times odd is odd.
In the exterior algebra of the plane, with basis $1, e_1, e_2, e_1 e_2$, the even part is spanned by $1$ and $e_1 e_2$ and the odd part by $e_1$ and $e_2$.
There $e_1 e_2 = -e_2 e_1$: two odd elements anticommute, which is the typical "super" sign rule.