Finite dimensional representations of quantum affine algebras

量子仿射代数的有限维表示

• 批准号: 15540023
• 负责人: NAKAJIMA Hiraku
• 金额: $ 1.86万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540023/
• 关键词: quantum affine algebras; instanton counting; モジュライ空間; 分配関数

In the joint work with Jonathan Beck, I gave an explicit description of the crystal base of the upper triangular subalgebra of the quantum enveloping algebra U_q for arbitrary affine Lie algebras. Using this description, we proved that arbitrary extremal weight modules can be embedded into tensor products of level 0 fundamental representations. This result is an extension of a result obtained by Beck and me for a symmetric affine Lie algebra. Furthermore, we proved Lusztig's conjecture on cells of the crystal base of U_q. (The paper was published in Duke.Math.)I wrote the C program for the algorithm to compute q-characters of finite dimensional representation of quantum affine algebras. I performed the computation for the E_8 case, with which the previous program cannot deal. But the final answer was not obtained.On the other hand, I together with Kota Yoshioka, studied Nekrasov's partition, which is the generating function of the equivariant fundamental classes of moduli spaces of ins … More tantons on R^4, which are simplest examples of quiver varieties. It has an explicit combinatorial description in terms of Young tableaux, which we use to perform the experimental calculation via MAPLE with a super computer. Based on the calculation, we conjectured that the partition function satisfies the blowup formula, which determine the partition function recursively. We then proved this conjecture theoretically. As an application, we proved Nekrasov's conjecture affirmatively, i.e., the pole of the partition function with respect to the parameter ε_1, ε_2 is equal to the Seiberg-Witten prepotential. (The paper will appear in Invent.Math.)We gave a series of lectures on instanton counting and Seiberg-Witten prepotential, for which we published a lecture note. In it, we studied the genus 1 part of the partition function, and showed that it coincides with what was expected by physists.We further studied the relation between the partition function and Donaldson invariants with Lothar Goettsche. Donalson invariants are defined as integration of natural cohomology classes on moduli spaces of instantons on a 4-manifold. When b_+=1, the invariants depends on the choice of a Riemannian metric. The difference of invariants with respect to two metrics is given by the wall-crossing formula. We proved that the wall-crossing formula can be expressed by Nekrasov's partition function for rank 2 case. This is proved by studing the torus action on the moduli spaces on toric surfaces. Less

在与Jonathan Beck的合作工作中，我给出了任意仿射李代数的量子包络代数U_q的上三角子代数的晶体基的显式描述。利用这种描述，我们证明了任意极值权模可以嵌入到0级基本表示的张量积中。这个结果是Beck和我关于对称仿射李代数的一个结果的推广。进一步证明了Lusztig关于U_q的晶基胞腔的猜想。(The论文发表在《杜克数学》上。）我编写了计算量子仿射代数有限维表示的q-特征标的算法的C程序。我对E_8情形进行了计算，这是以前的程序所不能处理的。另一方面，我和科塔吉冈一起研究了Ins的模空间的等变基本类的生成函数Nekrasov划分 ...更多信息 R^4上的坦顿，它们是最简单的双簇例子。它有一个明确的组合描述的杨tableaux，我们用它来执行实验计算通过MAPLE与超级计算机。在计算的基础上，我们证明了配分函数满足爆破公式，该公式递归地确定了配分函数。然后我们从理论上证明了这个猜想。作为应用，我们肯定地证明了Nekrasov猜想，即，配分函数关于参数ε_1，ε_2的极点等于Seiberg-Witten预势。(The论文将出现在Invent. Math中。）我们做了一系列关于瞬子计数和塞伯格-威滕预势的讲座，并为此发表了一份讲义。其中，我们研究了配分函数的亏格1部分，并证明了它与物理学家的预期是一致的，我们进一步研究了配分函数与Lothar Goettsche的唐纳森不变量之间的关系。Donalson不变量定义为4-流形上瞬子模空间上自然上同调类的积分。当B_+=1时，不变量依赖于黎曼度量的选择。关于两个度量的不变量的差由跨壁公式给出。证明了对于秩2的情况，越壁公式可以用Nekrasov配分函数来表示。通过研究复曲面上的模空间上的环面作用，证明了这一点。少

项目成果

t-Analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras

量子仿射代数的 Kirillov-Reshetikhin 模块的 q 字符的 t 类似物

• 发表时间: 2003
• 期刊: Representation Theory 7
• 作者: Nobushige Kurokawa;Hiroyuki Ochiai;H.Nakajima
• 通讯作者: H.Nakajima

Cells in quantum affine algebras

量子仿射代数中的元胞

• 发表时间: 2004
• 期刊: Algebra Colloquium 11・1
• 作者: Jonathan Beck;Hiraku Nakajima;Takao Watanabe;Hiraku Nakajima
• 通讯作者: Hiraku Nakajima

Convolution on homology groups of moduli spaces of sheaves on K3 Surfaces

K3面上滑轮模空间同调群的卷积

• 发表时间: 2003
• 期刊: Contemp.Math. 322
• 作者: Hiraku Nakajima;Kota Yoshioka;R.Coulangeon;Hiraku Nakajima;T.Watanabe;Hiraku Nakajima;T.Watanabe;Hiraku Nakajima
• 通讯作者: Hiraku Nakajima

Hiraku Nakajima: "Quiver varieties and t-analogs of q-characters of quantum affine algebras"Ann.of Math.. (to appear).

Hiraku Nakajima：“量子仿射代数的 q 字符的 Quiver 簇和 t 类似物”Ann.of Math..（即将出现）。

Hiraku Nakajima: "Geometric construction of representations of affine algebras"Proceedings of the International Congress of Mathematicians, Vol.I(Beijing,2002). 423-438

Hiraku Nakajima：“仿射代数表示的几何构造”，国际数学家大会论文集，第一卷（北京，2002）。