Heisenberg double of the generalized quantum euclidean group and its representations

TL;DR

This paper explores the structure and representations of the Heisenberg double of the generalized quantum Euclidean group, focusing on weight modules, prime spectrum, and automorphisms.

Contribution

It introduces the Heisenberg double of the generalized quantum Euclidean group and analyzes its modules, spectrum, and automorphism group, extending prior work on quantum groups.

Findings

Characterization of weight modules for the Heisenberg double

Description of the prime spectrum of the algebra

Determination of the automorphism group

Abstract

The generalized quantum Euclidean group $\oq(\frak{b}_{m,n})$ is a natural generalization of the quantum Euclidean group $\oq(\frak{b}_{1,1})$. The Heisenberg double $\od(\frak{b}_{m,n})$ of $\oq(\frak{b}_{m,n})$ is the smash product of $\oq(\frak{b}_{m,n})$ with its Hopf dual $\ou(\frak{b}_{m,n})$. In this paper, we study the weight modules, the prime spectrum and the automorphism group of the Heisenberg double $\od(\frak{b}_{m,n})$.