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

Abstract

Using the fusion product of the representations of the Lie algebra ${\mathfrak{sl}}_2$ we construct a set of the integrable highest weight $\widehat{{\mathfrak{sl}}_2}$-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 $\widehat{{\mathfrak{sl}}_2}$-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 $L^D$.

22 pages