The centrally extended Heisenberg double of quantum ${\mathrm{SL}}_2$

Abstract

The centrally extended Heisenberg double of quantum ${\mathrm{SL}}_2$ is a smash product of the quantized coordinate ring ${\mathcal{O}}_q(M_2)$ of $2\times 2$ matrices and the Hopf algebra $U_q({\mathfrak{sl}}_2)$. For this algebra, we determine its centre and show that it satisfies the quantum Gelfand–Kirillov conjecture. We give an explicit description of its prime, primitive and maximal ideals. We study a class of non-weight modules known as quasi-Whittaker modules. We classify the simple quasi-Whittaker modules and determine the annihilator of each simple quasi-Whittaker module.