The Heisenberg double of the quantum Euclidean group and its representations

TL;DR

This paper studies the algebraic structure and representations of the Heisenberg double of the quantum Euclidean group, providing explicit spectra descriptions, automorphism groups, and classifying simple modules.

Contribution

It offers the first explicit descriptions of spectra, automorphisms, and a classification of simple modules for the Heisenberg double of the quantum Euclidean group.

Findings

All prime factors are generalized Weyl algebras.

The algebra has no finite-dimensional representations.

It cannot have a Hopf algebra structure.

Abstract

The Heisenberg double $D_q(E_2)$ of the quantum Euclidean group $\mathcal{O}_q(E_2)$ is the smash product of $\mathcal{O}_q(E_2)$ with its Hopf dual $U_q(\mathfrak{e}_2)$. For the algebra $D_q(E_2)$, explicit descriptions of its prime, primitive, and maximal spectra are obtained. All prime factors of $D_q(E_2)$ are presented as generalized Weyl algebras. As a result, we obtain that the algebra $D_q(E_2)$ has no finite-dimensional representations, and that $D_q(E_2)$ cannot have a Hopf algebra structure. The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined. Some centralizers are explicitly described via generators and defining relations. This enables us to give a classification of simple weight modules, and the so-called $a$-weight modules, over the algebra $D_q(E_2)$.