On realizations of nonlinear Lie algebras by differential operators

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.

21 pages, Latex; New examples added in Sect. 3

References in corpus (2)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Degeneracy Structure of the Calogero-Sutherland Model: an Algebraic Approach

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• Polynomial algebras and exact solutions of general quantum non-linear optical models II: Multi-mode boson systems
• Landau quantum systems: an approach based on symmetry
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Nonlinear supersymmetry as a hidden symmetry
• The Involutive System of Higher-Spin Equations
• Conformal bridge in a cosmic string background
• Polynomial deformations of $sl(2)$ and unified algebraic framework for solutions of a class of spin models