The prime spectrum of the algebra \(\mathbb{K}_q[X,Y] \rtimes U_q(\mathfrak{sl}_2)\) and a classification of simple weight modules
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)].
scientific article; zbMATH DE number 7003042
scientific article