Quantum Heisenberg Group and Algebra: Contraction, Left and Right Regular Representations

TL;DR

This paper explores the quantum Heisenberg group and algebra, showing their relationship to quantum $SU_q(2)$, and constructs their regular representations, including irreducible and coherent state representations.

Contribution

It introduces the contraction method from quantum $SU_q(2)$ to the quantum Heisenberg group and derives explicit regular representations and their reductions.

Findings

Quantum $H_q(1)$ obtained from quantum $SU_q(2)$ via contraction.

Explicit left and right regular representations constructed.

Reduction yields finite-dimensional irreducible representations.

Abstract

We show that the quantum Heisenberg group $H_{q}(1)$ can be obtained by means of contraction from quantum $SU_q(2)$ group. Its dual Hopf algebra is the quantum Heisenberg algebra $U_{q}(h(1))$. We derive left and right regular representations for $U_{q}(h(1))$ as acting on its dual $H_{q}(1)$. Imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.