Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

Abstract

Parabosonic algebra in finite or infinite degrees of freedom is considered as a -graded associative algebra, and is shown to be a -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 . The bosonization technique for switching a Hopf algebra in the braided monoidal category (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper, we prove that for the parabosonic algebra PB, beyond the application of the bosonization technique to the original super-Hopf algebra, a bosonization-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra PB to an ordinary Hopf algebra, thus producing two different variants of PB, with an ordinary Hopf structure.