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Study on the path models for extremal weight modules over a quantum affine algebra

量子仿射代数极值权模路径模型研究

基本信息

批准号：
14540006
负责人：
NAITO Satoshi
金额：
$ 2.56万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540006/
关键词：
crystal base quantum affine algebra path model extremal weight module standard module Lakshmibai-Seshadri path level zero fundamental representation

项目摘要

M.Kashiwara introduced the notion of an extremal weight module $V(lambda)$ with an integral weight $lambda$ as an extremal weight over the quantum group $U_q(g)$ associated to a (general) Kac-Moody algebra $g$, by generalizing the notion of an integrable highest weight module. When $g$ is an affine Lie algebra and an integral weight $lambda$ is of level zero, we obtain certain finite-dimensional representations of a quantum affine algebra as quotient modules of the extremal weight module $V(lambda)$. For example, we obtain standard modules $M(lambda)$ introduced by H.Nakajima by an algebro-geometric method, which have turned out to coincide with quantum Weyl modules $M(lambda)$ introduced by V.Chari and A.Pressley. In our series of works from 2002 to 2004, for an integral weight $lambda$ of level zero that is a sum of level-zero fundamental weights for the affine Lie algebra $g$, we studied a certain crystal $B(lambda)_{cl}$, which is (modulo the null root of $g$) the crystal of all Lakshmibai-Seshadri paths of shape $lambda$. As a result, we proved that this crystal $B(lambda)_{cl}$ is isomorphic as a crystal to a tensor product of the path models for level-zero fundamental representations of the quantum affine algebra. From this result, we see that the crystal $B(lambda)_{cl}$ can be thought of as a path model for the standard module $M(lambda)$. Moreover, using our results above, we can describe the branching rule of the standard module $M(lambda)$ over the quantum affine algebra with respect to the restriction to the quantum group $U_q(g_0)$ associated to a (canonical) finite-dimensional reductive Lie subalgebra $g_0$ of $g$ in a combinatorial way, by using elements of the Weyl group $W$ of $g$ and the Bruhat order on $W$.

M.Kashiwara通过推广可积最高权模的概念，引入了量子群U_q（g）上的极值权模V（lambda）的概念，其中极值权模V（lambda）的极值权为整数权。当g是仿射李代数且整权λ是零级时，我们得到了量子仿射代数的某些有限维表示作为极权模V（λ）的商模.例如，我们用代数几何方法得到了H.Nakajima引入的标准模M（lambda），它与V.Chari和A. Pressley引入的量子Weyl模M（lambda）是一致的。在我们2002年至2004年的系列工作中，对于仿射李代数g$的零级基本权之和的零级整数权$lambda$，我们研究了一个晶体$B（lambda）_{cl}$，它是（模g$的零根）所有形状为$lambda$的Lakshmibai-Seshadri路的晶体。作为结果，我们证明了该晶体$B（lambda）_{cl}$作为晶体同构于量子仿射代数的零级基本表示的路径模型的张量积。从这个结果，我们看到晶体$B（lambda）_{cl}$可以被认为是标准模块$M（lambda）$的路径模型。此外，利用上述结果，我们还可以利用g的Weyl群W的元素和W上的Bruhat序，以组合的方式描述量子仿射代数上标准模M（lambda）关于量子群U_q（g）的分支规则，其中量子群U_q（g）与g的（正则）有限维约化李子代数g_0相关联.