Ladder operators, Fock-spaces, irreducibility and group gradings for the Relative Parabose Set algebra

The Fock-like representations of the Relative Parabose Set (\textsc{Rpbs}) algebra in a single parabosonic and a single parafermionic degree of freedom are investigated. It is shown that there is an infinite family (parametrized by the values of a positive integer $p$) of infinite dimensional, non-equivalent, irreducible representations. For each one of them, explicit expressions are computed for the action of the generators and they are shown to be ladder operators (creation-annihilation operators) on the specified Fock-spaces. It is proved that each one of these inf. dim. Fock-spaces is irreducible under the action of the whole algebra or in other words that it is a simple module over the \textsc{Rpbs} algebra. Finally, $(\mathbb{Z}_{2} \times \mathbb{Z}_{2})$-gradings are introduced for both the algebra $P_{BF}^{(1,1)}$ and the Fock-spaces, the constructed representations are shown to be $(\mathbb{Z}_{2} \times \mathbb{Z}_{2})$-graded, $P_{BF}^{(1,1)}$-modules and the relation between our present approach and similar works in the literature is briefly discussed.

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