The paraboson Fock space and unitary irreducible representations of the Lie superalgebra osp(1|2n)

TL;DR

This paper constructs explicit unitary irreducible representations of the Lie superalgebra osp(1|2n) using paraboson operators, providing matrix elements, basis vectors, and character formulas, which were previously difficult to obtain.

Contribution

It offers a novel explicit construction of the representations V(p) of osp(1|2n), including their matrix elements and character formulas, overcoming previous computational challenges.

Findings

Explicit construction of V(p) representations of osp(1|2n)

Derivation of character formulas for these representations

Explicit realization of representations of sp(2n) via branching

Abstract

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this implies that the ``parabosons of order p'' correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching $ osp(1|2n) \supset sp(2n) \supset u(n)$, we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters.