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

TL;DR

This paper explores a new Z_2 x Z_2-graded Lie superalgebra pso(2m+1|2n), analyzing its subalgebra structure and constructing associated parastatistics Fock spaces with explicit operator actions.

Contribution

It introduces and studies the structure of the novel pso(2m+1|2n) algebra, providing explicit constructions of its Fock spaces and basis labeling.

Findings

Constructed the Fock spaces for pso(2m+1|2n)

Provided explicit basis and operator actions

Identified the algebra's subalgebra structure

Abstract

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.