Solutions of the compatibility conditions for a Wigner quantum oscillator

Abstract

We consider the compatibility conditions for a N-particle D-dimensional Wigner quantum oscillator. These conditions can be rewritten as certain triple relations involving anticommutators, so it is natural to look for solutions in terms of Lie superalgebras. In the recent classification of ``generalized quantum statistics'' for the basic classical Lie superalgebras [math-ph/0504013], each such statistics is characterized by a set of creation and annihilation operators plus a set of triple relations. In the present letter, we investigate which cases of this classification also lead to solutions of the compatibility conditions. Our analysis yields some known solutions and several classes of new solutions.

9 pages

References in corpus (6)

• The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator
• Jacobson generators, Fock representations and statistics of sl(n+1)
• A non-commutative n-particle 3D Wigner quantum oscillator
• Macroscopic properties of A-statistics
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Cited by in corpus (4)

• Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System
• On the algebra of symmetries of Laplace and Dirac operators
• Representations of the Lie Superalgebra gl(1|n) in a Gel'fand-Zetlin Basis and Wigner Quantum Oscillators
• A classification of generalized quantum statistics associated with the exceptional Lie (super)algebras