Quantum counterparts of VII$_{a}$, III$_{a=1}$, VI$_{a\neq1}$ over harmonic oscillator

TL;DR

This paper constructs quantum analogs of specific three-dimensional Lie algebras using operadic Lax representations for the harmonic oscillator and analyzes their Jacobi operators in the semiclassical limit.

Contribution

It introduces a novel method to derive quantum versions of Lie algebras via operadic Lax representations, expanding the understanding of quantum algebra structures.

Findings

Quantum counterparts of VII$_{a}$, III$_{a=1}$, VI$_{a eq1}$ Lie algebras constructed

Jacobi operators studied in semiclassical approximation

New insights into quantum algebra properties obtained

Abstract

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some real three dimensional Lie algebras. The Jacobi operators of these quantum algebras are studied in semiclassical approximation.