Extremal weight modules of quantum affine algebras

Hiraku Nakajima

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

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

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