The parity operator for parafermions and parabosons

In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator P which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of n parafermions together with P is the orthogonal Lie algebra so(2n+2). The Fock spaces correspond to particular irreducible representations of so(2n+2), and the action of P in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of n parabosons together with P is the orthosymplectic Lie superalgebra osp(2|2n). In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of osp(2|2n). Both for parafermions and parabosons the spectrum of P is closely related to the so-called order of statistics p, introduced by Green.