On homomorphisms between global Weyl modules

Abstract:

Let $\mathfrak g$ be a simple finite-dimensional Lie algebra and let $A$ be a commutative associative algebra with unity. Global Weyl modules for the generalized loop algebra $\mathfrak g\otimes A$ were defined by Chari and Pressley (2001) and Feigin and Loktev (2004) for any dominant integral weight $\lambda$ of $\mathfrak g$ by generators and relations and further studied by Chari, Fourier, and Khandai (2010). They are expected to play a role similar to that of Verma modules in the study of categories of representations of $\mathfrak g\otimes A$. 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 global Weyl modules. This is done under certain restrictions on $\mathfrak g$, $\lambda$ and $A$. A crucial tool is the construction of fundamental global Weyl modules in terms of fundamental local Weyl modules.