$q$-Inverting pairs of linear transformations and the $q$-tetrahedron algebra

TL;DR

This paper introduces $q$-inverting pairs of linear transformations and establishes a bijection with irreducible modules of the $q$-tetrahedron algebra, advancing understanding of its representation theory.

Contribution

It defines $q$-inverting pairs and proves a bijection with irreducible $oxtimes_q$-modules, linking algebraic structures to module classification.

Findings

Defined $q$-inverting pairs of linear transformations.

Established a bijection with irreducible $oxtimes_q$-modules.

Provided a new framework for understanding the $q$-tetrahedron algebra.

Abstract

As part of our study of the $q$-tetrahedron algebra $\boxtimes_q$ we introduce the notion of a $q$-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each of which acts on the eigenspaces of the other according to a certain rule. Our main result is a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible $\boxtimes_q$-modules of type 1; (ii) the isomorphism classes of $q$-inverting pairs.