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.

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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