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

When the relative commutation relations between a set of m parafermions and n parabosons are of "relative parafermion type", the underlying algebraic structure is the classical orthosymplectic Lie superalgebra osp(2m+1|2n). The relative commutation relations can also be chosen differently, of "relative paraboson type". In this second case, the underlying algebraic structure is no longer 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 algebraic structure was performed by Tolstoy, amongst others. In the present paper, we investigate the subalgebra structure of pso(2m+1|2n). This allows us to study the parastatistics Fock spaces for this new set of m+n para-operators, as they correspond to lowest weight representations of pso(2m+1|2n). Our main result is the construction of these Fock spaces, with a complete labeling of the basis vectors and an explicit action of the para-operators on these basis vectors.