The $Z_2 \times Z_2$-graded Lie superalgebras $pso(2n+1|2n)$ and $pso(\infty|\infty)$, and parastatistics Fock spaces

Abstract: The parastatistics Fock spaces of order $p$ corresponding to an infinite number of parafermions and parabosons with relative paraboson relations are constructed. The Fock spaces are lowest weight representations of the $Z_2 \times Z_2$-graded Lie superalgebra $pso(\infty|\infty)$, with a basis consisting of row-stable Gelfand-Zetlin patterns.