On tensor products of irreducible integrable representations

TL;DR

This paper characterizes when tensor products of irreducible integrable representations of certain Borcherds--Kac--Moody algebras are isomorphic, extending classical results on tensor product factorization from finite-dimensional Lie algebras.

Contribution

It provides necessary and sufficient conditions for tensor product isomorphisms in a broad class of Borcherds--Kac--Moody algebra representations, generalizing known finite-dimensional Lie algebra results.

Findings

Derived conditions for tensor product isomorphisms

Extended classical factorization results to Borcherds--Kac--Moody algebras

Unified understanding of representation tensor products

Abstract

We consider integrable category $\mathcal{O}$ representations of Borcherds--Kac--Moody algebras whose Cartan matrix is finite dimensional, and determine the necessary and sufficient conditions for which the tensor product of irreducible representations from this category is isomorphic to another. This result generalizes a fundamental result of C. S. Rajan on unique factorization of tensor products of finite dimensional irreducible representations of finite dimensional simple Lie algebras over complex numbers.