The prime spectrum of the algebra \(\mathbb{K}_q[X,Y] \rtimes U_q(\mathfrak{sl}_2)\) and a classification of simple weight modules
type: Item
instance of: scholarly article
title: The prime spectrum of the algebra \(\mathbb{K}_q[X,Y] \rtimes U_q(\mathfrak{sl}_2)\) and a classification of simple weight modules
zbMATH Open document ID: 1408.17003
DOI: 10.4171/JNCG/294
published in: Journal of Noncommutative Geometry
publication date: 2019-01-17T00:00:00Z
review text: Let $\mathbb{K} K$ be a field. One classical object is the following smash product algebra $A:=\mathbb{K} K_q[X,Y]\rtimes U_q(\mathfrak{sl}_2)$, where $\mathbb{K} K_q[X,Y]:=\mathbb{K} K\langle X,Y| XY=qYX\rangle$ is the \textit{quantum plane}, $q\in\mathbb{K} K^*$ is not a root of unity, and $U_q(\mathfrak {sl}_2)$ is the quantized enveloping algebra of $\mathfrak{sl}_2$. As an abstract algebra, $A$ is generated over $\mathbb{K} K$ by elements $E,F,K,K^{-1}$, $X$, $Y$ subject to the defining relations ($K^{-1}$ is the inverse of $K$): \[ KEK^{-1} =q^2 E,\quad KFK^{-1}=q^{-2} F ,\quad [E,F]=\frac{K-K^{-1}}{q-q^{-1}}, \] \[ EX=qXE,\quad EY=X+q^{-1} YE, \quad FX=YK^{-1}+XF,\quad FY=YF, \] \[ KXK^{-1} =qX,\quad KYK^{-1}=q^{-1}Y,\quad qYX=XY. \] The authors show that the center of $A$ is generated by an explicit quadratic element. Explicit descriptions are given of the prime, primitive and maximal spectra of $A$. 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). \par The algebra $A$ can be seen as a quantum analogue of the enveloping algebra $U(V_2\rtimes \mathfrak{sl}_2)$ of the semidirect product Lie algebra $V_2\rtimes \mathfrak{sl}_2$, where $V_2$ is the 2-dimensional simple $\mathfrak{sl}_2$-module. The latter algebra was studied by the present authors in [J. Lie Theory 28, No. 2, 525--560 (2018; Zbl 1396.17011)].
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P50
zbMATH DE Number: 7003042
zbMATH Keywords: prime ideal
zbMATH Keywords: primitive ideal
zbMATH Keywords: weight module
zbMATH Keywords: simple module
zbMATH Keywords: centralizer
author: V. V. Bavula
author: Tao Lü
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7003042