The prime spectrum of the algebra Kq[X,Y]⋊Uq(sl2) and a classification of simple weight modules

© 2018 European Mathematical Society Publishing House. All rights reserved. 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 CA.K/of the quantum Cartan element K which is given in the paper. Explicit generators and defining relations are found for the algebra CA.K/(it is generated by 5 elements subject to the defining relations two of which are quadratic and one is cubic).