Khovanov Algebras for the Periplectic Lie Superalgebras

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 [ ]. We also give a diagrammatic description of the translation functors from [ ] 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.