Transformation Groups

, Volume 15, Issue 3, pp 517–549

A categorical approach to Weyl modules

  • Vyjayanthi Chari
  • Ghislain Fourier
  • Tanusree Khandai

DOI: 10.1007/s00031-010-9090-9

Cite this article as:
Chari, V., Fourier, G. & Khandai, T. Transformation Groups (2010) 15: 517. doi:10.1007/s00031-010-9090-9


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.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Vyjayanthi Chari
    • 1
  • Ghislain Fourier
    • 2
  • Tanusree Khandai
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany
  3. 3.Harish-Chandra Research InstituteAllahabadIndia