Advanced Studies in Pure Mathematics

Extremal weight modules of quantum affine algebras

Hiraku Nakajima

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Abstract

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra, and let $\mathbf{U}_q (\widehat{\mathfrak{g}})$ be the quantum affine algebra introduced by Drinfeld and Jimbo. In [11] Kashiwara introduced a $\mathbf{U}_q (\widehat{\mathfrak{g}})$-module $V(\lambda)$, having a global crystal base for an integrable weight $\lambda$ of level 0. We call it an extremal weight module. It is isomorphic to the Weyl module introduced by Chari-Pressley [6]. In [12, §13] Kashiwara gave a conjecture on the structure of extremal weight modules. We prove his conjecture when $\widehat{\mathfrak{g}}$ is an untwisted affine Lie algebra of a simple Lie algebra $\mathfrak{g}$ of type $ADE$, using a result of Beck-Chari-Pressley [5]. As a by-product, we also show that the extremal weight module is isomorphic to a universal standard module, defined via quiver varieties by the author [16, 18]. This result was conjectured by Varagnolo-Vasserot [19] and Chari-Pressley [6] in a less precise form. Furthermore, we give a characterization of global crystal bases by an almost orthogonality propery, as in the case of global crystal base of highest weight modules.

Article information

Source
Representation Theory of Algebraic Groups and Quantum Groups, T. Shoji, M. Kashiwara, N. Kawanaka, G. Lusztig and K. Shinoda, eds. (Tokyo: Mathematical Society of Japan, 2004), 343-369

Dates
Received: 9 March 2002
Revised: 16 October 2002
First available in Project Euclid: 3 January 2019

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1546542392

Digital Object Identifier
doi:10.2969/aspm/04010343

Mathematical Reviews number (MathSciNet)
MR2074599

Zentralblatt MATH identifier
1088.17008

Citation

Nakajima, Hiraku. Extremal weight modules of quantum affine algebras. Representation Theory of Algebraic Groups and Quantum Groups, 343--369, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04010343. https://projecteuclid.org/euclid.aspm/1546542392


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