# The prime spectrum of the algebra $K_{q}[X,Y]⋊U_{q}(sl_{2})$ and a classification of simple weight modules

### Vladimir V. Bavula

University of Sheffield, UK### Tao Lu

Huaqiao University, Quanzhou, Fujian, China

## 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).

## Cite this article

Vladimir V. Bavula, Tao Lu, The prime spectrum of the algebra $K_{q}[X,Y]⋊U_{q}(sl_{2})$ and a classification of simple weight modules. J. Noncommut. Geom. 12 (2018), no. 3, pp. 889–946

DOI 10.4171/JNCG/294