The parastatistics Fock space and explicit Lie superalgebra representations

Abstract

It is known that the defining triple relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations can be considered as defining relations for the Lie superalgebra osp(2m+1|2n) in terms of 2(m+n) generators. With the common Hermiticity conditions, this means that the parastatistics Fock space of order p corresponds to an infinite-dimensional unitary irreducible representation V(p) of osp(2m+1|2n), with lowest weight (-p/2,...,- p/|p/2,...,p/2). These representations (also in the simplest case m=n=1) had never been constructed due to computational difficulties, despite their importance. In the present paper we solve partially the problem in the general case using group theoretical techniques, in which the u(m|n) subalgebra of osp(2m+1|2n) plays a crucial role: a set of Gelfand-Zetlin patterns of u(m|n) can be used to label the basis vectors of V(p). An explicit and elegant construction of these representations V(p) for m=n=1, and the actions or matrix elements of the osp(3|2) generators are given.

References in corpus (6)

• The parafermion Fock space and explicit so(2n+1) representations
• Paraboson quotients. A braided look at Green ansatz and a generalization
• Gel'fand-Zetlin Basis and Clebsch-Gordan Coefficients for Covariant Representations of the Lie superalgebra gl(m|n)
• Graded Fock--like representations for a system of algebraically interacting paraparticles
• Ladder operators, Fock-spaces, irreducibility and group gradings for the Relative Parabose Set algebra
• Interpretation and Extension of the Green's Ansatz for Paraparticles

Cited by in corpus (4)

• Once more on parastatistics
• A class of infinite-dimensional representations of the Lie superalgebra osp(2m+1|2n) and the parastatistics Fock space
• Reduced matrix elements of the orthosymplectic Lie superalgebra
• Generalized Quantum Statistics and Lie (Super)Algebras