Solutions of the compatibility conditions for a Wigner quantum oscillator

TL;DR

This paper explores solutions to the compatibility conditions of a Wigner quantum oscillator using Lie superalgebras, identifying both known and new classes of solutions based on generalized quantum statistics.

Contribution

It connects the compatibility conditions with Lie superalgebra classifications, providing new solutions and insights into quantum oscillator models.

Findings

Identified known solutions to the compatibility conditions.

Discovered several new classes of solutions.

Linked solutions to generalized quantum statistics.

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.