A note on parabolic Verma module homomorphisms over Kac-Moody Algebras

TL;DR

This paper introduces a reduction rule for hom-spaces between parabolic Verma modules over different Kac-Moody algebras, advancing understanding of module homomorphisms in this algebraic context.

Contribution

It presents a novel reduction rule for hom-spaces between parabolic Verma modules across different Kac-Moody algebras, with applications to module homomorphisms.

Findings

Reduction rule for hom-spaces established

Applications to homomorphism classification demonstrated

Enhanced understanding of parabolic Verma modules provided

Abstract

We consider a general parabolic category $\mathcal{O}_{S}$ over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on parabolic Verma module homomorphisms.