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

Abstract

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.

14 pages, Work done during postdoctoral stay of the first author, at IFM, UMSNH, Morelia, Michoacan, Mexico. v2 contains detailed proofs for some intermediate formulas and corrections of some typos in the theorem of section 2 and in some formulas of section 3 as well. v3 corresponds to the final form accepted for publication at the International Journal of Algebra. To appear at spring-summer 2011

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