The Braided Heisenberg Group

TL;DR

This paper explores the structure of the braided Heisenberg group, its duality with quantum groups, and applications to braided harmonic oscillators, revealing new algebraic and physical insights.

Contribution

It computes braided groups for the quantum Heisenberg solution, establishes duality with the universal enveloping algebra, and applies these to braided harmonic oscillators.

Findings

Braided groups and matrices for the quantum Heisenberg solution are computed.

The braided Heisenberg group is shown to be self-dual.

An isomorphism between braided and unbraided tensor products is established.

Abstract

We compute the braided groups and braided matrices $B(R)$ for the solution $R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra $U_{q}(h)$, and use this result to derive an action of $U_{q}(h)$ on the braided groups. We then demonstrate the various covariance properties using the braided Heisenberg group as an explicit example. In addition, the braided Heisenberg group is found to be self-dual. Finally, we discuss a physical application to a system of n braided harmonic oscillators. An isomorphism is found between the n-fold braided and unbraided tensor products, and the usual `free' time evolution is shown to be equivalent to an action of a primitive generator of $U_{q}(h)$ on the braided tensor product.