A family of representations of the affine Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$

TL;DR

This paper constructs a new family of irreducible representations for the affine Lie superalgebra , extending Wakimoto's free field approach and identifying conditions for irreducibility.

Contribution

It introduces a new class of irreducible representations for , extending existing methods to the affine superalgebra context.

Findings

Representations are irreducible iff the parameter  is nonzero.

Constructed representations over  and .

Extended Wakimoto free field construction to affine Lie superalgebras.

Abstract

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $\mu$, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter $\mu$ is nonzero.