Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map

TL;DR

This paper demonstrates that parabosonic and parafermionic algebras possess a Hopf algebra structure, which can generate Lie algebra structures via a generalized Jordan Schwinger map, enriching the algebraic framework of paraparticles.

Contribution

It introduces a Hopf algebraic structure for parabosonic and parafermionic algebras and extends the Jordan Schwinger map to paraparticles, connecting these algebras to Lie algebra structures.

Findings

Hopf structure exists for parabosonic and parafermionic algebras

Hopf structure can generate Lie algebra representations

Discussion of differences between Hopf and graded Hopf superalgebra structures

Abstract

The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well known Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed.