On Homomorphisms Between Global Weyl Modules

TL;DR

This paper investigates the morphism structure of global Weyl modules for generalized loop algebras, establishing an analogue of Verma modules' homomorphism properties under specific conditions.

Contribution

It proves that the space of homomorphisms between global Weyl modules is either zero or one-dimensional, extending properties known for Verma modules.

Findings

Hom space between global Weyl modules is at most one-dimensional.

Non-zero homomorphisms are injective under certain conditions.

Construction of fundamental global Weyl modules from local ones.

Abstract

Global Weyl modules for generalized loop algebras $\lie g\tensor A$, where $\lie g$ is a simple finite dimensional Lie algebra and A is a commutative associative algebra were defined, for any dominant integral weight $\lambda$, by generators and relations. They are expected to play the role similar to that of Verma modules in the study of categories of representations of these algebras. One of the fundamental properties of Verma modules is that the space of morphisms between two Verma modules is either zero or one--dimensional and also that any non--zero morphism is injective. The aim of this paper is to establish an analogue of this property for the global Weyl modules. This is done under certain restrictions on the Lie algebra $\lie g$, $\lambda$ and $A$. A crucial tool is the construction of fundamental global Weyl modules in terms of fundamental local Weyl modules given in Section 3.