Super-Hopf realizations of Lie Superalgebras: Braided Paraparticle extensions of the Jordan-Schwinger map

TL;DR

This paper explores the algebraic structure of mixed paraparticle systems, introduces new Lie superalgebra realizations with super-Hopf properties, and discusses applications in representation theory and theoretical physics.

Contribution

It presents a novel family of Lie superalgebra realizations that preserve super-Hopf structures and applies braided group theory to paraparticle systems.

Findings

Braided group structure for the Relative Parabose Set.

New realizations of Lie superalgebras with super-Hopf transfer.

Potential applications in representation theory and Skyrme model solutions.

Abstract

The mathematical structure of a mixed paraparticle system (combining both parabosonic and parafermionic degrees of freedom) commonly known as the Relative Parabose Set, will be investigated and a braided group structure will be described for it. A new family of realizations of an arbitrary Lie superalgebra will be presented and it will be shown that these realizations possess the valuable representation-theoretic property of transferring invariably the super-Hopf structure. Finally two classes of virtual applications will be outlined: The first is of interest for both mathematics and mathematical physics and deals with the representation theory of infinite dimensional Lie superalgebras, while the second is of interest in theoretical physics and has to do with attempts to determine specific classes of solutions of the Skyrme model.