Representations of parafermi rings

Abstract

We consider the operator algebra corresponding to a system consisting of an arbitrary finite number v of parafermi oscillators with a view to obtaining all irrepresentations of the operator algebra defining it. It is shown that this algebra is isomorphic to the Lie algebra of the orthogonal group B_v in 2v+1 dimensions. The known results of the representation theory of the orthogonal group are then used to find the representations of the ring of parafermi operators. In particular it is seen that the so-called Green ansatz for a parafermi ring arises in a natural way and that, in a certain sense, it furnishes the most general "solution" of the parafermi algebra. The case of a parafermi ring of order 2, which is the simplest non-trivial case, is considered in some detail.