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

27 pages, some typos of the journal version are corrected and a couple of references added

References in corpus (2)

• Paraboson quotients. A braided look at Green ansatz and a generalization
• Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map