The Braided Heisenberg Group

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.

33 pages

Cited by in corpus (6)

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• Hopf algebra gauge theory on a ribbon graph
• Braided Oscillators
• Quantum Group Covariance and the Braided Structure of Deformed Oscillators
• Modified toric code models with flux attachment from Hopf algebra gauge theory
• Generalized exclusion and Hopf algebras