Evaluation modules for the $q$-tetrahedron algebra

TL;DR

This paper provides a detailed classification and explicit matrix representations of evaluation modules for the $q$-tetrahedron algebra, including basis transformations and automorphism actions, enriching the understanding of its module structure.

Contribution

It offers a comprehensive description of evaluation modules for $oxtimes_q$, including bases, matrix representations, transition matrices, and automorphism effects, which was not previously detailed.

Findings

24 attractive bases for evaluation modules are identified.

Explicit matrices for algebra generators are provided for each basis.

Automorphisms via $ ext{Z}_4$ are characterized and related to module twisting.

Abstract

Let $\mathbb F$ denote an algebraically closed field, and fix a nonzero $q \in \mathbb F$ that is not a root of unity. We consider the $q$-tetrahedron algebra $\boxtimes_q$ over $\mathbb F$. It is known that each finite-dimensional irreducible $\boxtimes_q$-module of type 1 is a tensor product of evaluation modules. This paper contains a comprehensive description of the evaluation modules for $\boxtimes_q$. This description includes the following topics. Given an evaluation module $V$ for $\boxtimes_q$, we display 24 bases for $V$ that we find attractive. For each basis we give the matrices that represent the $\boxtimes_q$-generators. We give the transition matrices between certain pairs of bases among the 24. It is known that the cyclic group $\Z_4$ acts on $\boxtimes_q$ as a group of automorphisms. We describe what happens when $V$ is twisted via an element of $\Z_4$. We discuss how evaluation modules for $\boxtimes_q$ are related to Leonard pairs of $q$-Racah type.