Quiver varieties and tensor products
Quiver varieties and tensor products
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DOI：
10.1007/pl00005810
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发表时间：
2001-11-01
影响因子：
3.1
通讯作者：
Nakajima, H
中科院分区：
文献类型：
--
作者：
Nakajima, H
本文利用张量簇给出了不同范畴张量积的几何构造。更精确地说，我们在一个代数簇中引入了一个波浪号上的拉格朗日子簇（3）。（1）（3）在波浪号上的同调群是对称Kac-Moody李代数g的一个表示，它同构于张量积V（lambda（1））圈乘（.）可积最高权模的圈乘V（lambda（N））。(2)（3）在波浪号上的不可约分量的集合具有晶体的结构，同构于V（lambda（1））圈乘以（.）的q-模拟的结构。循环乘以V（lambda（N））。(3)当g是ADE型时，（3）的等变K-同调群同构于量子圈代数U-q（Lg）的泛标准模的张量积.我们也给出了（2）的晶体的纯组合描述。这个结果即使在N = 1时也是新的。
In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety (3) over tilde in a quiver variety. and show the following results: (1) The homology group of (3) over tilde is a representation of a symmetric Kac-Moody Lie algebra g, isomorphic to the tensor product V(lambda (1)) circle times (...)circle times V(lambda (N)) of integrable highest weight modules. (2) The set of irreducible components of (3) over tilde has a structure of a crystal, isomorphic to that of the q-analogue of V(lambda (1)) circle times (...)circle times V(lambda (N)). (3) The equivariant K-homology group of (3 ) over tilde is isomorphic to the tensor product of universal standard modules of the quantum loop algebra U-q(Lg), when g is of type ADE. We also give a purely combinatorial description of the crystal of (2). This result is new even when N = 1.