On realizations of nonlinear Lie algebras by differential operators

TL;DR

This paper explores how polynomial deformations of the sl(2,R) Lie algebra can be represented using differential operators linked to bosonic operators, analyzing their finite and infinite-dimensional representations with applications in quantum optics.

Contribution

It introduces a general method for realizing nonlinear Lie algebras of arbitrary degree via differential operators related to bosonic operators, with explicit cases for linear, quadratic, and cubic deformations.

Findings

Explicit realizations for linear, quadratic, and cubic cases

Method applicable to arbitrary polynomial degrees

Applications demonstrated in quantum optics

Abstract

We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are studied in the context of these ``nonlinear'' Lie algebras and some examples dealing with quantum optics are pointed out.