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When the relative commutation relations between a set of m parafermions and nparabosons are of ``relative parafermion type'', the underlying algebraicstructure is the classical orthosymplectic Lie superalgebra osp(2m+1|2n). Therelative commutation relations can also be chosen differently, of ``relativeparaboson type''. In this second case, the underlying algebraic structure is nolonger an ordinary Lie superalgebra, but a Z_2 x Z_2$-graded Lie superalgebra,denoted here by pso(2m+1|2n). The identification of this new algebraicstructure was performed by Tolstoy, amongst others. In the present paper, weinvestigate the subalgebra structure of pso(2m+1|2n). This allows us to studythe parastatistics Fock spaces for this new set of m+n para-operators, as theycorrespond to lowest weight representations of pso(2m+1|2n). Our main result isthe construction of these Fock spaces, with a complete labeling of the basisvectors and an explicit action of the para-operators on these basis vectors.

The Z_2 x Z_2-graded Lie superalgebra pso(2m+1|2n) and new parastatistics representations