Quantum algebras and Lie groups

TL;DR

This paper explores the connection between Lie groups and quantum algebras via their universal enveloping algebras, extending group actions to quantum structures, and demonstrating physical applications to phonon representations.

Contribution

It introduces a method to handle quantum structures similarly to Lie groups without topological complications, and applies this to phonon representations in quantum physics.

Findings

Quantum structures can be treated as Lie groups via their universal enveloping algebras.

The adjoint action extends naturally to q-algebras and q-coalgebras.

Application to phonons demonstrates group action and fusion using coproducts.

Abstract

Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can be dealt more or less as the Lie one and we do not need to introduce the not easy to handle topological groups. Composed system also is described by the suitably symmetrized q-coalgebra. A physical application to the phonon, irreducible unitary representation of E_q(1,1), shows both the transformation under the group action of one phonon state and the fusion of two phonons, by means of the coproduct, in only one phonon lying on a branch of the appropriate dispersion relation.