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 $osp(2\mathrm{\mid }2n)$. In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of $osp(2\mathrm{\mid }2n)$. Both for parafermions and parabosons the spectrum of $P$ is closely related to the so-called order of statistics $p$, introduced by Green.