Orthosymplectic $Z_2\times Z_2$-graded Lie superalgebras and parastatistics

A $Z_2\times Z_2$-graded Lie superalgebra $g$ is a $Z_2\times Z_2$-graded algebra with a bracket $[\mathrm{.},\mathrm{.}]$ 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 $Z_2\times Z_2$-graded Lie superalgebra $osp(2m_1+1,2m_2\mathrm{\mid }2n_1,2n_2)$ 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 $Z_2\times Z_2$-graded matrix. Since the orthosymplectic Lie superalgebra $osp(2m+1\mathrm{\mid }2n)$ is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from $osp(2m_1+1,2m_2\mathrm{\mid }2n_1,2n_2)$. Some special cases are of particular interest, even when one is dealing with parabosons only.