Work

Khovanov Algebras for the Periplectic Lie Superalgebras

Year: 2024

Type: article

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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnae230

Author: Jonas Nehme

Institution: Max Planck Institute for Mathematics

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,534

Funders: Deutsche Forschungsgemeinschaft, International Max Planck Research School for Advanced Methods in Process and Systems Engineering, Hausdorff Center for Mathematics