Polynomial deformations of $osp(1/2)$ and generalized parabosons

TL;DR

This paper explores polynomial deformations of the superalgebra $osp(1/2)$, classifies their irreducible representations for quadratic cases, and links these structures to generalized paraboson algebras and deformed oscillator algebras.

Contribution

It introduces polynomial deformations of $osp(1/2)$, classifies irreducible representations for degrees up to two, and connects these deformations to generalized parabosons.

Findings

Complete classification for degree ≤ 2 polynomial deformations.

Identification of similarities and differences with classical $osp(1/2)$ representations.

Interpretation of the algebra as a generalized paraboson algebra.

Abstract

We consider the algebra $R$ generated by three elements $A,B,H$ subject to three relations $[H,A]=A$, $[H,B]=-B$ and $\{A,B\}=f(H)$. When $f(H)=H$ this coincides with the Lie superalgebra $osp(1/2)$; when $f$ is a polynomial we speak of polynomial deformations of $osp(1/2)$. Irreducible representations of $R$ are described, and in the case $\deg(f)\leq 2$ we obtain a complete classification, showing some similarities but also some interesting differences with the usual $osp(1/2)$ representations. The relation with deformed oscillator algebras is discussed, leading to the interpretation of $R$ as a generalized paraboson algebra.