Graded Fock--like representations for a system of algebraically interacting paraparticles

TL;DR

This paper introduces a new algebraic framework for mixed paraparticle systems, focusing on Fock-like representations with a parameter p, and explores their mathematical properties including ladder operators, irreducibility, and gradings.

Contribution

It presents the first detailed study of Fock-like representations for the Relative Parabose Set algebra with a focus on the parameter p and their structural properties.

Findings

Construction of ladder operators for the representations

Proof of irreducibility of the modules

Establishment of Klein group gradings in the algebra

Abstract

We will present an algebra describing a mixed paraparticle model, known in the bibliography as "The Relative Parabose Set (\textsc{Rpbs})". Focusing in the special case of a single parabosonic and a single parafermionic degree of freedom $P_{BF}^{(1,1)}$, we will study a class of Fock--like representations of this algebra, dependent on a positive integer parameter p (a kind of generalized parastatistics order). Mathematical properties of the Fock--like modules will be investigated for all values of p and constructions such as ladder operators, irreducibility (for the carrier spaces) and Klein group gradings (for both the carrier spaces and the algebra itself) will be established.