Duflo–Serganova functor and superdimension formula for the periplectic Lie superalgebra

Inna Entova-Aizenbud, Vera Serganova

Research output: Contribution to journalComment/debate

Abstract

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple p(n)-module L and a certain odd element x ∈ p(n) of rank 1, we give an explicit description of the composition factors of the p(n−1)-module DSx (L), which is defined as the homology of the complex ∏Mx−→ Mx−→ ∏M, where ∏ denotes the parity-change functor (−) ⊗ ℂ0|1. In particular, we show that this p(n−1)-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional p(n)-module, based on its highest weight. In particular, this reproves the Kac–Wakimoto conjecture for p(n), which was proved earlier by the authors.

Original languageEnglish
Pages (from-to)697-729
Number of pages33
JournalAlgebra and Number Theory
Volume16
Issue number3
DOIs
StatePublished - 1 Jan 2022

Keywords

  • Duflo–Serganova functor
  • Lie superalgebra
  • periplectic Lie superalgebra
  • superdimension

ASJC Scopus subject areas

  • Algebra and Number Theory

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