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, that is, 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 [ 5]. We also give a diagrammatic description of the translation functors from [ 1] 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 $\operatorname{Ext}^{1}$ between irreducible modules and show that $\mathfrak{p}(n)\operatorname{-mod}$ does not admit a Koszul grading.