Variants of bosonisation in Parabosonic algebra. The Hopf and super-Hopf structures

TL;DR

This paper demonstrates that parabosonic algebra can be structured as a super-Hopf algebra and introduces two methods, including bosonisation, to convert it into an ordinary Hopf algebra, revealing new algebraic variants.

Contribution

It establishes the super-Hopf algebra structure of parabosonic algebra directly and introduces two distinct bosonisation-like techniques to obtain ordinary Hopf algebra variants.

Findings

Parabosonic algebra is a $bZ_2$-graded super-Hopf algebra.

Two different methods successfully convert the super-Hopf algebra into an ordinary Hopf algebra.

The techniques produce two algebraic variants with different structures.

Abstract

Parabosonic algebra in finite or infinite degrees of freedom is considered as a $\mathbb{Z}_{2}$-graded associative algebra, and is shown to be a $\mathbb{Z}_{2}$-graded (or: super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the $osp(1/2n)$ Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$. The bosonisation technique for switching a Hopf algebra in the braided monoidal category ${}_{H}\mathcal{M}$ (where $H$ is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper we prove that for the parabosonic algebra $P_{B}$, beyond the application of the bosonisation technique to the original super-Hopf algebra, a bosonisation-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra $P_{B}$ to an ordinary Hopf algebra, producing thus two different variants of $P_{B}$, with ordinary Hopf structure.