Quiver varieties and quantum affine algebras

箭袋簇和量子仿射代数

• 批准号: 13640019
• 负责人: NAKAJIMA Hiraku
• 金额: $ 1.54万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640019/
• 关键词: Quiver varieties; Quantum affine algebra; vector bundles over K3 surfaces; moduli spaces; K3曲面の上のベクトル束; 結晶基底; ラグランジアン部分多様体; テンソル積

Standard modules, introduced by the head investigator via equivariant K-groups of quiver varieties, are proved to be isomorphic to extremal weight modules, introduced by Kashiwara. It was shown by Kashiwara that extremal weight modules have crystal bases. The head investigator proved that they are 'almost orthonormal' with respect to the natural inner product. This result is generalized to arbitrary afiine Lie algebras by a joint work with J. Beck. As an application, we prove the conjecture of Lusztig on cells of quantum affine algebras.On the other hand, the head investigator further studies t-analogs of q-characters. In particular, he gives 1) expressions in terms of Young tableaux for type A and D, and 2) generalization to the case when q is a root of unity. He also proved that the q-characters of a certain class of representations, called Kirillov-Reshetkhin modules, satisfy the recursive system called T-system.Also, the head investigator writes a C program for computing q-characters of finite dimensional representations of quantum affine algebras. He succeeded the calculation except Eg case. For Eg case, he can succeed if he has enough memory (some ten giga bytes) and computer time (about one week). But he did not have enough budget to perform the computation.The head investigator also studies operators on cohomology groups of moduli spaces of vector bundles over K3 surfaces, given by exceptional vector bundles.

标准模，由首席研究员通过等变K-群的K-品种，被证明是同构的极值重量模，介绍了柏原。Kashiwara证明了极值权模有晶体基。首席研究员证明，他们是“几乎正交”的自然内积。与J. Beck的合作将这一结果推广到了任意线性李代数。作为应用，我们证明了Lusztig关于量子仿射代数胞腔的猜想，并进一步研究了q-特征标的t-类似。特别是，他给出了1）关于类型A和D的Young tableaux的表达式，以及2）当q是单位根时的推广。他还证明了某类表示（称为Kirillov-Reshetkhin模）的q-特征标满足称为T-系统的递归系统。此外，首席研究员编写了计算量子仿射代数有限维表示的q-特征标的C程序。除例外，他的计算都成功了。例如，如果他有足够的内存（大约10千兆字节）和计算机时间（大约一周），他就可以成功。但他没有足够的预算来执行计算。首席调查员还研究运营商上同调群的模空间的向量丛在K3表面，给出了例外的向量丛。

项目成果

Hiraku Nakajima: "t-analogs of q-characters of quantum affine of type An. Dn"Cont. Math.. (to appear).

Hiraku Nakajima：“An.Dn 类型量子仿射的 q 字符的 t 类似物”续。

Hiraku Nakajima: "Extremal weight modules of quantum affine algebras"Advanced Studies in Pure Mathematics, Representation Theory of Algebraic Groups. (to appear).

Hiraku Nakajima：《量子仿射代数的极值权模》纯数学高级研究，代数群表示论。

Hiraku Nakajima: "t-analogs of q-characters of quantum affine algebras of type A_n, D_n"Cont.Math.. (to appear).

Hiraku Nakajima：“A_n、D_n 型量子仿射代数的 q 字符的 t 类似物”Cont.Math..（即将出现）。

Hiraku Nakajima: "Geometric constructions of representations of affine algebras"Proc. of Int. Congr. of Math.. vol. 1. 423-438 (2003)

Hiraku Nakajima：“仿射代数表示的几何构造”Proc。

Hiraku Nakajima: "Cells in quautum affine algebras"Proc. of Int. Conf. on Algebras Suzhou. (to appear).

Hiraku Nakajima：“量子仿射代数中的细胞”Proc。