Verma modules and finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity

TL;DR

This paper classifies finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity, showing they are quotients of Verma modules and providing a complete classification.

Contribution

It extends the classification of irreducible modules of the universal Askey--Wilson algebra to roots of unity, identifying all such modules as quotients of Verma modules.

Findings

Finite-dimensional irreducible modules with marginal weights are quotients of Verma modules.

Two families of quotients account for all such irreducible modules.

Complete classification of these modules up to isomorphism.

Abstract

Assume that $\mathbb F$ is an algebraically closed field and fix a nonzero scalar $q\in \mathbb F$ with $q^4\not=1$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative algebra over $\mathbb F$ defined by generators and relations. The generators are $A,B,C$ and the relations assert that each of \begin{gather*} A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}} \end{gather*} commutes with $A,B,C$. The Verma $\triangle_q$-modules are a family of infinite-dimensional $\triangle_q$-modules with marginal weights. Under the condition that $q$ is not a root of unity, it was shown that every finite-dimensional irreducible $\triangle_q$-module has a marginal weight and is isomorphic to a quotient of a Verma $\triangle_q$-module. Assume that $q$ is a root of unity. We prove that every finite-dimensional irreducible $\triangle_q$-module with a marginal weight is isomorphic to a quotient of a Verma $\triangle_q$-module. Properly speaking, two natural families of finite-dimensional quotients of Verma $\triangle_q$-modules contain all finite-dimensional irreducible $\triangle_q$-modules with marginal weights up to isomorphism. Furthermore, we classify the finite-dimensional irreducible $\triangle_q$-modules with marginal weights up to isomorphism.