Advanced Studies in Pure Mathematics

Extremal weight modules of quantum affine algebras

Hiraku Nakajima

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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

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

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

Permanent link to this document euclid.aspm/1546542392

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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.

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