Grothendieck rings for Lie superalgebras and the Duflo–Serganova functor

Hoyt, Crystal; Reif, Shifra

doi:10.2140/ant.2018.12.2167

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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

DOI: 10.2140/ant.2018.12.2167

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

Vol. 12, No. 9, 2018