Dynamical deformations of three-dimensional Lie algebras in Bianchi classification over the harmonic oscillator

TL;DR

This paper uses operadic Lax representations of the harmonic oscillator to construct and analyze dynamical deformations of 3D real Lie algebras in the Bianchi classification, linking energy conservation to algebraic identities.

Contribution

It introduces a method to generate dynamical deformations of Lie algebras via harmonic oscillator representations, proving these deformations preserve Lie algebra structure.

Findings

Dynamical deformations are shown to be Lie algebras.

Energy conservation relates to Jacobi identities.

Deformations depend on harmonic oscillator dynamics.

Abstract

Operadic Lax representations for the harmonic oscillator are used to construct the dynamical deformations of three-dimensional (3D) real Lie algebras in the Bianchi classification. It is shown that the energy conservation of the harmonic oscillator is related to the Jacobi identities of the dynamically deformed algebras. Based on this observation, it is proved that the dynamical deformations of 3D real Lie algebras in the Bianchi classification over the harmonic oscillator are Lie algebras.