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

Entova-Aizenbud, Inna; Serganova, Vera

doi:10.2140/ant.2022.16.697

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

Inna Entova-Aizenbud and Vera Serganova

Vol. 16 (2022), No. 3, 697–729

DOI: 10.2140/ant.2022.16.697

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 $x \in 𝔭 (n)$ of rank $1$ , we give an explicit description of the composition factors of the $𝔭 (n - 1)$ -module ${DS}_{x} (L)$ , which is defined as the homology of the complex

Π M \to_{}^{x} M \to_{}^{x} Π M,

where $Π$ denotes the parity-change functor $(-) \otimes ℂ^{0 | 1}$ .

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.

To Pavel Etingof for his 50th birthday.

Keywords

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

Mathematical Subject Classification

Primary: 17B10, 17B55

Milestones

Received: 13 April 2020

Revised: 13 June 2021

Accepted: 24 July 2021

Published: 9 July 2022

Authors

Inna Entova-Aizenbud
Department of Mathematics Ben-Gurion University Beer-Sheva Israel

Vera Serganova
Department of Mathematics University of California at Berkeley Berkeley, CA United States

Vol. 16, No. 3, 2022