Vol. 12, No. 9, 2018

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Grothendieck rings for Lie superalgebras and the Duflo–Serganova functor

Crystal Hoyt and Shifra Reif

Vol. 12 (2018), No. 9, 2167–2184
Abstract

We show that the Duflo–Serganova functor on the category of finite-dimensional modules over a finite-dimensional contragredient Lie superalgebra induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the ring of supercharacters. We realize this homomorphism as a certain evaluation of functions related to the supersymmetry property. We use this realization to describe the kernel and image of the homomorphism induced by the Duflo–Serganova functor.

Keywords
Lie superalgebra, supercharacter, Grothendieck ring, Duflo–Serganova functor, supersymmetric Laurent polynomials
Mathematical Subject Classification 2010
Primary: 17B10
Secondary: 05E05, 05E10
Milestones
Received: 8 October 2017
Revised: 1 June 2018
Accepted: 20 July 2018
Published: 21 December 2018
Authors
Crystal Hoyt
Department of Mathematics
Weizmann Institute of Science
ORT Braude College
Rehovot
Israel
Shifra Reif
Department of Mathematics
Bar-Ilan University
Ramat-Gan
Israel