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

Variants of bosonisation in Parabosonic algebra. The Hopf and super-Hopf structures - Mathematical Physics - Download this document for free, or read online. Document in PDF available to download.

Abstract: Parabosonic algebra in finite or infinite degrees of freedom is considered asa $\mathbb{Z} {2}$-graded associative algebra, and is shown to be a$\mathbb{Z} {2}$-graded or: super Hopf algebra. The super-Hopf algebraicstructure of the parabosonic algebra is established directly without appealingto its relation to the $osp1-2n$ Lie superalgebraic structure. The notion ofsuper-Hopf algebra is equivalently described as a Hopf algebra in the braidedmonoidal category ${} {\mathbb{CZ} {2}}\mathcal{M}$. The bosonisation techniquefor switching a Hopf algebra in the braided monoidal category${} {H}\mathcal{M}$ where $H$ is a quasitriangular Hopf algebra into anordinary Hopf algebra is reviewed. In this paper we prove that for theparabosonic algebra $P {B}$, beyond the application of the bosonisationtechnique to the original super-Hopf algebra, a bosonisation-like constructionis also achieved using two operators, related to the parabosonic total numberoperator. Both techniques switch the same super-Hopf algebra $P {B}$ to anordinary Hopf algebra, producing thus two different variants of $P {B}$, withordinary Hopf structure.

Author: ** K. Kanakoglou, C. Daskaloyannis**

Source: https://arxiv.org/