On a class of Fock-like representations for Lie Superalgebras

TL;DR

This paper develops a new family of infinite-dimensional Fock-like super-representations for Lie superalgebras using realizations via the Relative Parabose Set algebra, expanding the understanding of their structure and decompositions.

Contribution

It introduces a novel class of Fock-like representations for Lie superalgebras parameterized by a positive integer, constructed through algebra realizations and subalgebra inclusions.

Findings

Constructed infinite-dimensional, decomposable super-representations for any Lie superalgebra.

Representations are parameterized by a positive integer p.

Applied to study decompositions with respect to various low-dimensional Lie algebras and superalgebras.

Abstract

Utilizing Lie superalgebra (LS) realizations via the Relative Parabose Set algebra $P_{BF}$, combined with earlier results on the Fock-like representations of $P_{BF}^{(1,1)}$, we proceed to the construction of a family of Fock-like representations of LSs: these are infinite dimensional, decomposable super-representations, which are parameterized by the value of a positive integer $p$. They can be constructed for any LS $L$, either initiating from a given 2-dimensional, $\mathbb{Z}_{2}$-graded representation of $L$ or using its inclusion as a subalgebra of $P_{BF}^{(1,1)}$. As an application we proceed in studying decompositions with respect to various low-dimensional Lie algebras and superalgebras.