Supermodule - Knowlpedia

Definition

Let $A=A_{\bar0}\oplus A_{\bar1}$ be a superalgebra. A left supermodule over $A$ is a super vector space $M=M_{\bar0}\oplus M_{\bar1}$ with a unital associative action satisfying

A_{\bar i}M_{\bar j}\subseteq M_{\bar i+\bar j}.

Equivalently, the action $A\otimes M\to M$ is an even map in $\mathbf{SuperVect}$ .

Morphisms and internal maps

Morphisms of supermodules are usually even $A$ -linear maps. A homogeneous linear map $f:M\to N$ of parity $\epsilon$ is internally $A$ -linear when

f(am)=(-1)^{\epsilon|a|}a\,f(m)

for homogeneous $a\in A$ . Thus degree-zero internal maps are precisely the ordinary morphisms, while degree-one internal maps are odd module maps.

Left and right modules

Koszul signs enter when one converts a left action into a right action. If $A$ is supercommutative and $M$ is a left $A$ -supermodule, the compatible right action is

m a=(-1)^{|m||a|}a m

on homogeneous elements. Omitting this sign generally breaks associativity with the super symmetry.

Clifford modules

Because a Clifford algebra is $\mathbb Z/2$ -graded, a graded Clifford module is exactly a supermodule over that superalgebra. An ungraded Clifford module instead forgets the parity information.

References

V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, American Mathematical Society, 2004. DOI record. Relevant: Chapter 1.
I. M. Musson, Lie Superalgebras and Enveloping Algebras, American Mathematical Society, 2012. DOI record. Relevant: Chapter 1.