A categorical approach to Weyl modules

By Vyjayanthi Chari, Ghislain Fourier, Tanusree Khandai. 2009

141 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f14f600b452106df7531a19a4b0aa3393b89c08

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Citations in the corpus

Representations of Multiloop Algebras (2008, methodology): “Since the release of earlier versions of this paper, our approach has already been adapted to classify modules for Lie algebras g⊗ A of g-valued functions on affine schemes Spec(A) and their invariants under more general finite group actions [Chari et al. 2009; Neher et al. 2009].”
Irreducible finite-dimensional representations of equivariant map algebras (2009, background): “ [CP86], [CP98]. Their work has had a long-lasting impact. Generalizations and more precise descriptions of their work have appeared in many papers, for example in Batra [Bat04], Chari-FourierKhandai [CFK], Chari-Fourier-Senesi[CFS08], Chari-Moura[CM04], Feigin-Loktev Received by the editors April 1”
Affine highest weight categories and affine quasihereditary algebras (2014, background): “Here we follow [8], see also [3, 4, 7, 9] and references therein.”
Demazure Modules, Fusion Products and Q-Systems (2013, background): “It was proved in [11] (see also [7]) that the local Weyl modules are finite–dimensional and so, in particular, we have (xα ⊗ 1) wλ = 0, α ∈ R .”
Weyl modules and $$q$$q-Whittaker functions (2012, unknown): “In addition, according to [3,12] there exists an action of C[Aλ̌] on W(λ̌) such that”