Orthosymplectic Z2 x Z2-graded Lie superalgebras and parastatistics

A Z2 x Z2-graded Lie superalgebra g is a Z2 x Z2-graded algebra with a bracket [·, ·] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z2 x Z2-graded Lie superalgebra osp(2m1+1, 2m2|2n1, 2n2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z2 x Z2-graded matrix. Since the orthosymplectic Lie superalgebra osp(2m + 1|2n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from osp(2m1+1, 2m2|2n1, 2n2). Some special cases are of particular interest, even when one is dealing with parabosons only.