Grothendieck rings for Lie superalgebras and the Duflo-Serganova functor

TL;DR

This paper explores how the Duflo-Serganova functor induces a ring homomorphism on Grothendieck rings of finite-dimensional modules over Lie superalgebras, linking it to character rings and supersymmetry properties.

Contribution

It establishes a ring homomorphism induced by the Duflo-Serganova functor on Grothendieck rings and characterizes its kernel and image using supersymmetry-related evaluations.

Findings

Ring homomorphism on Grothendieck ring induced by Duflo-Serganova functor

Explicit description of the kernel and image of the homomorphism

Connection between the homomorphism and supersymmetry properties

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 characters. 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.