Harmonic Oscillator Lie Bialgebras and their Quantization

TL;DR

This paper classifies all Lie bialgebra structures on the harmonic oscillator algebra, introduces a new quantum oscillator via quantization of a triangular Lie bialgebra, and provides a universal R-matrix for this quantum algebra.

Contribution

It explicitly derives all Lie bialgebra structures on the harmonic oscillator algebra and constructs a new quantum oscillator with an associated universal R-matrix.

Findings

All Lie bialgebra structures are of the coboundary type.

A new quantum oscillator is introduced as a quantization of a triangular Lie bialgebra.

A universal R-matrix for the new quantum algebra is provided.

Abstract

All possible Lie bialgebra structures on the harmonic oscillator algebra are explicitly derived and it is shown that all of them are of the coboundary type. A non-standard quantum oscillator is introduced as a quantization of a triangular Lie bialgebra, and a universal $R$-matrix linked to this new quantum algebra is presented.