Unique factorization of tensor products for Kac-Moody Algebras [HBNI Th55]

Abstract

In the first part, we address a fundamental question, unique factorization of tensor products, that arises in representation theory. 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. In the second part, we get a new interpretation of the chromatic polynomials using Kac-Moody theory and derive some of its properties using this new interpretation.