The Prime Spectrum and Representation Theory of the $2\times 2$ Reflection Equation Algebra

TL;DR

This paper investigates the prime spectrum and representation theory of the 2x2 reflection equation algebra using generalized Weyl algebras, classifying modules, automorphisms, and prime ideals, and establishing the Dixmier-Moeglin equivalence.

Contribution

It provides a detailed analysis of the prime spectrum, module classification, and automorphisms of the algebra, and proves the Dixmier-Moeglin equivalence for it.

Findings

Classification of simple finite-dimensional modules

Automorphism group computation

Prime spectrum with Zariski topology and Dixmier-Moeglin equivalence

Abstract

The theory of generalized Weyl algebras is used to study the $2\times 2$ reflection equation algebra $\mathcal{A}=\mathcal{A}_q(\operatorname{M}_2)$ in the case that $q$ is not a root of unity, where the $R$-matrix used to define $\mathcal{A}$ is the standard one of type $A$. Simple finite dimensional $\mathcal{A}$-modules are classified, finite dimensional weight modules are shown to be semisimple, $\operatorname{Aut}(\mathcal{A})$ is computed, and the prime spectrum of $\mathcal{A}$ is computed along with its Zariski topology. Finally, it is shown that $\mathcal{A}$ satisfies the Dixmier-Moeglin equivalence.