# Representations of the Lie superalgebra $B(\infty,\infty)$ and parastatistics Fock spaces

## TL;DR

This paper extends the algebraic framework of parafermions and parabosons to infinite systems, developing new matrix forms and bases for the Lie superalgebra B(∞,∞) and its Fock spaces.

## Contribution

It introduces a new matrix form and Gelfand-Zetlin basis for B(n,n), enabling analysis of the infinite-dimensional case B(∞,∞) and its Fock space representations.

## Key findings

- Constructed a new matrix form for B(n,n)
- Developed a Gelfand-Zetlin basis for finite cases
- Extended structures to infinite rank B(∞,∞)

## Abstract

The algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is known to be the Lie superalgebra osp(2m+1|2n). The Fock spaces of such systems are then certain lowest weight representations of osp(2m+1|2n). In the current paper, we investigate what happens when the number of parafermions and parabosons becomes infinite. In order to analyze the algebraic structure, and the Fock spaces, we first need to develop a new matrix form for the Lie superalgebra B(n,n)=osp(2n+1|2n), and construct a new Gelfand-Zetlin basis of the Fock spaces in the finite rank case. The new structures are appropriate for the situation $n\rightarrow\infty$. The algebra generated by the infinite number of creation and annihilation operators is $B(\infty,\infty)$, a well defined infinite rank version of the orthosymplectic Lie superalgebra. The Fock spaces are lowest weight representations of $B(\infty,\infty)$, with a basis consisting of particular row-stable Gelfand-Zetlin patterns.