# Journal of Noncommutative Geometry

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**Volume 12, Issue 3, 2018, pp. 889–946**

**DOI: 10.4171/JNCG/294**

Published online: 2018-10-29

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

Vladimir V. Bavula^{[1]}and Tao Lu

^{[2]}(1) University of Sheffield, UK

(2) Huaqiao University, Quanzhou, Fujian, China

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).

*Keywords: *Prime ideal, primitive ideal, weight module, simple module, centralizer

Bavula Vladimir, Lu Tao: The prime spectrum of the algebra $\mathbb K_q[X,Y]\rtimes U_q(\mathfrak {sl}_2)$ and a classification of simple weight modules. *J. Noncommut. Geom.* 12 (2018), 889-946. doi: 10.4171/JNCG/294