The affine VW supercategory

Abstract

We define the affine VW supercategory , which arises from studying the action of the periplectic Lie superalgebra $$\mathfrak {p}(n)$$ on the tensor product $$M\otimes V^{\otimes a}$$ of an arbitrary representation M with several copies of the vector representation V of $$\mathfrak {p}(n)$$. It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in $$\mathfrak {p}(n)\otimes \mathfrak {p}(n)$$. When M is the trivial representation, the action factors through the Brauer supercategory $$\textit{s}\mathcal {B}{} \textit{r}$$. Our main result is an explicit basis theorem for the morphism spaces of and, as a consequence, of $$\textit{s}\mathcal {B}{} \textit{r}$$. The proof utilises the close connection with the representation theory of $$\mathfrak {p}(n)$$. As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.