Supercommutative algebra

Definition

Let $k$ be a field of characteristic different from $2$ . A superalgebra $A$ is supercommutative when its homogeneous elements satisfy

ab=(-1)^{|a||b|}ba.

Equivalently, its multiplication is unchanged by the Koszul braiding in $\mathbf{SuperVect}_k$ , so $A$ is a commutative algebra object in the category of super vector spaces.

Consequence for odd elements

If $a$ is odd, supercommutativity gives $a^2=-a^2$ . Because $2$ is invertible in $k$ , every odd element therefore satisfies

a^2=0.

This conclusion need not follow in characteristic $2$ ; definitions intended for that setting commonly impose additional conditions on odd squares.

Basic examples

The exterior algebra $\Lambda W$ , graded by exterior degree modulo $2$ , is supercommutative. More generally,

\operatorname{Sym}(U)\otimes\Lambda(W)

is supercommutative when $U$ is even and $W$ is odd. These algebras are the local coordinate models in smooth and algebraic supergeometry.

By contrast, a Clifford algebra is naturally a superalgebra but is generally not supercommutative: its odd generators can have nonzero squares.

Terminology warning

“Graded-commutative” is also used for $\mathbb Z$ -graded algebras with sign $(-1)^{ij}$ . Reducing a $\mathbb Z$ -grading modulo $2$ gives the present rule, but it forgets the integer degree.

References

C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, European Mathematical Society, 2011. DOI record. Relevant: Chapter 1.
Y. I. Manin, Gauge Field Theory and Complex Geometry, second edition, Springer, 1997. DOI record. Relevant: Chapter 4.