Symmetric algebra of a super vector space

Core idea

Let $k$ be a field of characteristic different from $2$ , and let $V$ be a super vector space. Its symmetric superalgebra is

\operatorname{Sym}_{\mathrm s}(V) =T(V)\Big/\left\langle v\otimes w-(-1)^{|v||w|}w\otimes v \right\rangle,

where the relations range over homogeneous $v,w$ and $T(V)$ is the tensor algebra. It is the free supercommutative algebra generated by $V$ .

Even and odd generators

Writing $V=V_{\bar0}\oplus V_{\bar1}$ , multiplication induces an isomorphism of superalgebras

\operatorname{Sym}_{\mathrm s}(V) \cong \operatorname{Sym}(V_{\bar0})\otimes\Lambda(V_{\bar1}).

Even generators commute and produce the ordinary symmetric algebra, whereas odd generators anticommute and produce the exterior algebra. In particular, the square of each odd generator vanishes.

Universal property

For every supercommutative algebra $A$ , restriction to $V$ gives a natural bijection

\operatorname{Hom}_{\mathrm{sCAlg}_k} (\operatorname{Sym}_{\mathrm s}(V),A) \cong \operatorname{Hom}_{\mathbf{SuperVect}_k}(V,A).

The maps on the right are even linear maps. This universal property, rather than a choice of bases, characterizes the construction.

Relation to Lie superalgebras

The symmetric superalgebra appears as the associated graded algebra in the Poincaré–Birkhoff–Witt theorem for the universal enveloping algebra of a Lie superalgebra.

References

I. M. Musson, Lie Superalgebras and Enveloping Algebras, American Mathematical Society, 2012. DOI record. Relevant: Chapters 1 and 6.
V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, American Mathematical Society, 2004. DOI record. Relevant: Chapter 1.