The Hahn superalgebra and supersymmetric Dunkl oscillator models

TL;DR

This paper introduces the Hahn superalgebra, a supersymmetric extension of the Hahn algebra, constructed via Dunkl oscillator models with reflections, and explores their supersymmetric properties and invariance in multiple dimensions.

Contribution

It presents the first construction of the Hahn superalgebra using Dunkl oscillators with reflections and develops supersymmetric Dunkl oscillator models in multiple dimensions.

Findings

The Hahn superalgebra is realized through Dunkl oscillators with reflection operators.

Supersymmetric Dunkl oscillator models in n dimensions are constructed with reflection-based Hamiltonians.

In two dimensions, the supersymmetric Dunkl oscillator's invariance algebra is the Hahn superalgebra.

Abstract

A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian involves reflection operators. In this realization, the reflections act as grading operators and the odd generators are part of the Schwinger-Dunkl algebra, which is a two-parameter extension of the bosonic su(2) construction. The even part of the algebra is built from bilinears in the odd generators and satisfy the Hahn algebra supplemented with involutions. A family of supersymmetric Dunkl oscillator models in n dimensions is also considered. The Hamiltonians of these supersymmetric models differ from ordinary Dunkl oscillators by pure reflection terms. In two dimensions, the supersymmetric Dunkl oscillator is seen to have the even part of the Hahn superalgebra as invariance algebra.