A Class of Representations of the Orthosymplectic Lie Superalgebras $$\mathcal{B}(n,n)$$ and $$\mathcal{B}(\infty ,\infty )$$

Abstract

In 1982 Palev showed that 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 the Lie superalgebra osp(2m + 1|2n). The "parastatistics Fock spaces" of order p of such systems are then certain lowest weight representations of osp(2m + 1|2n). We investigate now the situation when the number of parafermions and parabosons becomes infinite, which is of interest not only in a physics context but also from the mathematical point of view. In this contribution, we will discuss the various steps that are needed to understand the infinite-rank case. First, we will introduce appropriate bases and Dynkin diagrams for B(n, n) = osp(2n + 1|2n) that allow us to extend n → ∞. Then we will develop a new matrix form for B(n, n) = osp(2n + 1|2n), because the standard one is not appropriate for taking this limit. Following this, we construct a new Gelfand-Zetlin basis of the parastatistics Fock spaces in the finite rank case (in correspondence with this new matrix form). The new structures, related to a non-distinguished simple root system, allow the extension to n → ∞. This leads to the definition of the algebra B(∞, ∞) as a Lie superalgebra generated by an infinite number of creation and annihilation operators (subject to certain relations), or as an algebra of certain infinite-dimensional matrices. We study the parastatistics Fock spaces, as certain lowest weight representations of B(∞, ∞). In particular, we construct a basis consisting of well-described row-stable Gelfand-Zetlin patterns.