Parabosons, parafermions and representations of ℤ₂ x ℤ₂-graded Lie superalgebras

For a set of m parafermion operators and n paraboson operators, there are two nontrivial ways to unify them in a larger algebraic structure. One of these corresponds to the orthosymplectic Lie superalgebra osp(2m + 1 vertical bar 2n). The other one is no longer a Z(2)-graded Lie superalgebra but a Z(2) x Z(2)-graded Lie superalgebra, a rather different algebraic structure, denoted here by pso(2m + 1 vertical bar 2n). In a recent paper, the Fock spaces (V) over tilde (p) of order p for pso(2m+1 vertical bar 2n) were determined. In the current paper, we summarize some of the main properties of pso(2m+1 vertical bar 2n) and its Fock spaces. In particular, we concentrate on the Fock space for p = 1, and indicate how it reduces to an ordinary boson-fermion Fock space