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

The theory of generalized Weyl algebras is used to study the $2\times 2$ reflection equation algebra $\mathcal{A}={\mathcal{A}}_q({\mathrm{M}}_2)$ 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, $\mathrm{Aut}(\mathcal{A})$ 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.