Khovanov algebras for the periplectic Lie superalgebras

Abstract

The periplectic Lie superalgebra $\mathfrak{p}(n)$ is one of the most mysterious and least understood simple classical Lie superalgebras with reductive even part. We approach the study of its finite dimensional representation theory in terms of Schur--Weyl duality. We provide an idempotent version of its centralizer, i.e. the super Brauer algebra. We use this to describe explicitly the endomorphism ring of a projective generator for $\mathfrak{p}(n)$ resembling the Khovanov algebra of [BS11a]. We also give a diagrammatic description of the translation functors from [BDE19] in terms of certain bimodules and study their effect on projective, standard, costandard and irreducible modules. These results will be used to classify irreducible summands in $V^{\otimes d}$, compute $\mathrm{Ext}^1$ between irreducible modules and show that $\mathfrak{p}(n)$-mod does not admit a Koszul grading.