Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

Journal Article · Mon Jul 15 04:00:00 EDT 2013 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4816086· OSTI ID:22218262

Marquette, Ian ^[1]

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1. School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072 (Australia)

We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.

• OSTI ID: 22218262
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 7 Vol. 54; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ALGEBRA
CASIMIR OPERATORS
ENERGY SPECTRA
INTEGRALS
IRREDUCIBLE REPRESENTATIONS
OSCILLATORS
POISSON EQUATION
POLYNOMIALS
STRUCTURE FUNCTIONS