Grothendieck Rings of Queer Lie Superalgebras

TL;DR

This paper characterizes the Grothendieck rings of finite-dimensional modules over Queer Lie superalgebras, revealing their structure via character rings and completing the classification for all classical Lie superalgebras.

Contribution

It explicitly determines the Grothendieck rings of Queer Lie superalgebras and describes their isomorphism to specific Laurent polynomial rings, completing the classification for classical Lie superalgebras.

Findings

Grothendieck ring of $Q(n)$ is isomorphic to Laurent polynomials with a specific evaluation property.

Complete description of Grothendieck rings for all classical Lie superalgebras.

Character ring structure is explicitly determined for Queer Lie supergroups.

Abstract

We determine the Grothendieck rings of the category of finite-dimensional modules over Queer Lie superalgebras via their rings of characters. In particular, we show that the ring of characters of the Queer Lie supergroup $Q(n)$ is isomorphic to the ring of Laurent polynomials in $x_{1},\ldots,x_{n}$ for which the evaluation $x_{n-1}=-x_{n}=t$ is independent of $t$. We thus complete the description of Grothendieck rings for all classical Lie superalgebras.