Vol. 16, No. 3, 2022

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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
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(n1)-module DS x(L), which is defined as the homology of the complex

ΠM xM xΠM,

where Π denotes the parity-change functor () 0|1.

In particular, we show that this 𝔭(n1)-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