Integrable $\hat{\mathfrak{sl}_2}$-modules as infinite tensor products

TL;DR

The paper constructs and analyzes a family of integrable highest weight modules for the affine Lie algebra fsl_2, using fusion products, providing bases, decompositions, and character formulas for these modules.

Contribution

It introduces a new family of modules for fsl_2fhat, constructed via fusion products, with explicit bases, decompositions, and character formulas, extending previous irreducible module classifications.

Findings

Constructed modules depend on vector D in bbN^{k+1}

Modules include irreducible cases for special D

Derived explicit bases, decompositions, and character formulas

Abstract

Using the fusion product of the representations of the Lie algebra $\mathfrak{sl}_2$ we construct a set of the integrable highest weight $\hat{\mathfrak{sl}_2}$-modules $L^D$, depending on the vector $D\in\mathbb{N}^{k+1}$. In a special cases of $D$ our modules are isomorphic to the irreducible $\hat{\mathfrak{sl}_2}$-modules $L_{i,k}$. We construct a basis of the $L^D$ and study the decomposition of $L^D$ on the irreducible components. We also write a formulas for the characters of $L^D$.