Finite dimensional representations of quantum affine algebras

在与Jonathan Beck的联合工作中，我给出了任意仿射李代数的量子包络代数U_q上三角子代数的晶体基的明确描述。利用这一描述，我们证明了任意极值权模块可以嵌入到0级基本表示的张量积中。这个结果是Beck和我对对称仿射李代数所得到的结果的推广。进一步，我们在U_q晶体基的胞上证明了Lusztig的猜想。（这篇论文发表在《杜克数学》杂志上。）我为计算量子仿射代数有限维表示的q字符的算法编写了C程序。我执行了E_8情况的计算，这是前面的程序无法处理的。但是最终的答案并没有得到。另一方面，我与Kota Yoshioka一起研究了Nekrasov划分，它是in的模空间的等变基本类的生成函数。R^4上的更多的子，这是最简单的颤变的例子。它有一个明确的Young表的组合描述，我们使用它在超级计算机上通过MAPLE进行实验计算。根据计算，我们推测配分函数满足爆破公式，该公式递归地确定配分函数。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，即配分函数对参数ε_1， ε_2的极点等于Seiberg-Witten预势。（这篇论文将发表在《发明数学》杂志上。）我们做了一系列关于瞬时计数和Seiberg-Witten预电位的讲座，并发表了一份课堂笔记。在这篇文章中，我们研究了配分函数的属1部分，并证明了它与物理学家所期望的一致。我们与Lothar Goettsche进一步研究了配分函数与唐纳森不变量的关系。Donalson不变量被定义为4流形上实例的模空间上的自然上同类的积分。当b_+=1时，不变量取决于黎曼度规的选择。关于两个度量的不变量之差由过墙公式给出。证明了在秩2的情况下，用Nekrasov配分函数表示过壁公式。通过研究环面对环面模空间的作用证明了这一点。少

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