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.

Figures (10)

• Figure 1: Different cases of the surgery procedure
• Figure 2: Pictorial description of the maps $\hat{\eta}_i$, $\hat{\epsilon}_i$ and $\hat{\psi}_{i,j}$
• Figure 3: How to interpret $\hat{\eta}_i$, $\hat{\epsilon}_i$ and $\hat{\psi}_{i,i+1}$ as rotated surgery procedures
• Figure 4: Example of how $\lambda'$ is constructed.
• Figure 5: Example for the first surgery procedure of $\underline{\lambda}\overline{\lambda'}\cdot\underline{\lambda'}\overline{\mu}$

Theorems & Definitions (194)

• Theorem A
• Theorem B
• Theorem C
• Theorem D: Main theorem
• Definition 1.1
• Remark 1.2
• Definition 1.3
• Lemma 1.4
• Corollary 1.5
• proof