The prime spectrum of the algebra ${\mathbb{K}}_q[X,Y]⋊U_q({\mathfrak{sl}}_2)$ and a classification of simple weight modules

Abstract

For the algebra $A$ in the title, it is shown that its centre is generated by an explicit quartic element. Explicit descriptions are given of the prime, primitive and maximal spectra of the algebra $A$. A classification of simple weight $A$-modules is obtained. The classification is based on a classification of (all) simple modules of the centralizer $C_A(K)$ of the quantum Cartan element $K$ which is given in the paper. Explicit generators and defining relations are found for the algebra $C_A(K)$ (it is generated by 5 elements subject to the defining relations two of which are quadratic and one is cubic).