Metrics related to Oscillator algebras

TL;DR

This paper explores the structural properties, invariant metrics, and potential extensions of oscillator Lie K-algebras, a class of solvable metric Lie algebras generalizing the harmonic oscillator algebra.

Contribution

It introduces oscillator Lie K-algebras as double extensions of metric spaces and analyzes their structural features and invariant metrics.

Findings

Identified structural features of oscillator Lie K-algebras

Described invariant metrics and derivations of these algebras

Explored possibilities for extending to mixed metric Lie algebras

Abstract

A Lie algebra is said to be metric if it admits a symmetric invariant and nondegenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable nonabelian metric example. This algebra is the first step of a countable series of solvable Lie algebras which support invariant Lorentzian forms. Generalizing this situation, in this paper we arrive to the oscillator Lie K-algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.