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Representation Theory
Representation Theory
ISSN 1088-4165

 

On homomorphisms between global Weyl modules


Authors: Matthew Bennett, Vyjayanthi Chari, Jacob Greenstein and Nathan Manning
Journal: Represent. Theory 15 (2011), 733-752
MSC (2010): Primary 17B10, 17B37
Published electronically: December 20, 2011
MathSciNet review: 2869017
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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.


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Additional Information

Matthew Bennett
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: mbenn002@math.ucr.edu

Vyjayanthi Chari
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: vyjayanthi.chari@ucr.edu

Jacob Greenstein
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: jacob.greenstein@ucr.edu

Nathan Manning
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: nmanning@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S1088-4165-2011-00407-6
PII: S 1088-4165(2011)00407-6
Received by editor(s): December 2, 2010
Received by editor(s) in revised form: March 9, 2011
Published electronically: December 20, 2011
Additional Notes: The second and third authors were partially supported by DMS-0901253 (V.C.) and DMS-0654421 (J.G.)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.