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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.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Riverside, CA, 92521, USA

    Vyjayanthi Chari

  2. Mathematisches Institut, Universität zu Köln, Köln, Germany

    Ghislain Fourier

  3. Harish-Chandra Research Institute, Allahabad, India

    Tanusree Khandai

Authors
  1. Vyjayanthi Chari
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  2. Ghislain Fourier
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  3. Tanusree Khandai
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Corresponding author

Correspondence to Vyjayanthi Chari.

Additional information

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)

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Chari, V., Fourier, G. & Khandai, T. A categorical approach to Weyl modules. Transformation Groups 15, 517–549 (2010). https://doi.org/10.1007/s00031-010-9090-9

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  • DOI: https://doi.org/10.1007/s00031-010-9090-9

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