Ladder Operators, Fock-Spaces, Irreducibility and Group Gradings for the Relative Parabose Set Algebra

The Fock-like representations of the Relative Parabose Set (Rpbs) algebra in a single parabosonic and a single parafermionic degree of freedom are investigated. It is shown that there is an innite family (parametrized by the values of a positive integer p) of innite 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 specied 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 Rpbs algebra. Finally, (Z2 Z2)-gradings are introduced for both the algebra P (1;1) BF