Evaluation modules for the $q$-tetrahedron algebra

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 ${⊠}_q$ over $\mathbb{F}$. It is known that each finite-dimensional irreducible ${⊠}_q$-module of type 1 is a tensor product of evaluation modules. This paper contains a comprehensive description of the evaluation modules for ${⊠}_q$. This description includes the following topics. Given an evaluation module $V$ for ${⊠}_q$, we display 24 bases for $V$ that we find attractive. For each basis we give the matrices that represent the ${⊠}_q$-generators. We give the transition matrices between certain pairs of bases among the 24. It is known that the cyclic group ${\mathbb{Z}}_4$ acts on ${⊠}_q$ as a group of automorphisms. We describe what happens when $V$ is twisted via an element of ${\mathbb{Z}}_4$. We discuss how evaluation modules for ${⊠}_q$ are related to Leonard pairs of $q$-Racah type.

62 pages

References in corpus (8)

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Cited by in corpus (2)

• Leonard pairs having zero-diagonal TD-TD form
• Leonard pairs having LB-TD form