, Volume 15, Issue 3, pp 517-549
Date: 09 Apr 2010

A categorical approach to Weyl modules

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Abstract

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.

Partially supported by the NSF grant DMS-0901253. (Vyjayanthi Chari)
Supported by the DFG-project “Kombinatorische Beschreibung von Macdonald und Kostka–Foulkes Polynomen”. (Ghislain Fourier)