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

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.
Authors:
 [1]
  1. School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072 (Australia)
Publication Date:
OSTI Identifier:
22218262
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 7; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ALGEBRA; CASIMIR OPERATORS; ENERGY SPECTRA; INTEGRALS; IRREDUCIBLE REPRESENTATIONS; OSCILLATORS; POISSON EQUATION; POLYNOMIALS; STRUCTURE FUNCTIONS