The affine VW supercategory

Abstract

We define the affine VW supercategory $s\bigvee \bigvee$, 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 $s\mathcal{B}r$. Our main result is an explicit basis theorem for the morphism spaces of $s\bigvee \bigvee$ and, as a consequence, of $s\mathcal{B}r$. 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.

35 pages

References in corpus (1)

• 2-Kac-Moody algebras