Unique factorization of tensor products for Kac-Moody algebras

TL;DR

This paper proves a unique factorization property for tensor products of irreducible modules in the category of integrable modules over symmetrizable Kac-Moody algebras, extending finite-dimensional Lie algebra results.

Contribution

It generalizes Rajan's theorem to Kac-Moody algebras and introduces a new proof method combining representation theory and combinatorics.

Findings

Tensor products have unique factorizations up to one-dimensional twists.

The proof applies to both finite and infinite-dimensional cases.

The approach uses Kac-Weyl character formula analysis.

Abstract

We consider integrable, category O-modules of indecomposable symmetrizable Kac-Moody algebras. We prove that unique factorization of tensor products of irreducible modules holds in this category, upto twisting by one dimensional modules. This generalizes a fundamental theorem of Rajan for finite dimensional simple Lie algebras over C. Our proof is new even for the finite dimensional case, and uses an interplay of representation theory and combinatorics to analyze the Kac-Weyl character formula.