Parafermions, parabosons and representations of so(\infty) and osp(1|\infty)

TL;DR

This paper explicitly constructs Fock spaces for parafermions and parabosons with infinitely many generators, linking them to representations of infinite-dimensional Lie algebras and superalgebras, and provides basis and character formulas.

Contribution

It provides a complete explicit construction of Fock spaces for infinite parafermion and paraboson algebras, connecting them to so() and osp(1|) representations.

Findings

Basis vectors labeled by stable Gelfand-Zetlin patterns

Explicit transformation formulas for basis vectors

Character formulas for the Fock space representations

Abstract

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(\infty) and of the Lie superalgebra osp(1|\infty). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.