Symplectic Reflection Algebras

主要研究人员：Victor Ginzburg建议编号：0303465机构：芝加哥大学摘要：辛反射代数本建议的主要主题是将非对易几何的一般机制应用于代数和几何中的具体问题。该提议在看似不相关的数学分支之间建立了几个意想不到的新联系，如非对易代数、代数几何、表示理论和物理。具体地说，给定一个有限维辛向量空间和该向量空间的一个有限辛自同构群，我们研究了相应的对称结构的辛分解和辛形变(包括交换和非交换)。通过Hochschild上同调，这给出了一种新的方法来研究orbilold的各种“弦”拓扑不变量。因此，该提议的主要思想之一是通过非对易几何的技巧来逼近奥布里奥德的代数几何。“辛反射代数”为实现这一思想提供了一个基本工具。系统地使用辛反射代数导致了传统的“交换”代数几何中的各种新结果，而这些结果已经不能用纯粹的“交换”方法来获得。该提案为研究开辟了广泛的独立方向。该提案的基本目标之一是将在一个数学分支中发展的方法和结果应用于完全不同的数学领域。例如，该提案涉及系统地将各种代数技术应用于几何和表示理论中的问题。因此，这项提议加强了数学“统一”的思想。这一建议不仅在数学的各个领域有深远的应用，而且在现代数学和理论物理，特别是在量子物理中也有应用。数学科学作为一个整体，对社会做出了深刻的贡献。目前的建议在促进不同学科之间的互动和增进科学理解方面大有裨益。

Principal Investigator: Victor Ginzburg proposal Number: 0303465Institution: University of ChicagoAbstract: Symplectic reflection algebras The main theme of the present proposal is applying the general machinery of non-commutative geometry to concrete problems in algebra and geometry. The proposal establishes several new unexpected links between seemingly unrelated branches of mathematics, such as noncommutative algebra, algebraic geometry, representation theory, and physics. Specifically, given a finite dimensional symplectic vector space and a finite group of symplectic automorphisms of that vector space, the authors study symplectic resolutions and symplectic deformations (both commutative and noncommutative) of the corresponding orbifold. This gives a new approach, through Hochschild cohomology, to various "stringy" topological invariants of the orbifold. Thus, one of the main ideas of the proposal is to approach the algebraic geometry of an orbifold via the techniques of noncommutative geometry. The "symplectic reflection algebras" provide a basic tool for implementing this idea. A systematic use of symplectic reflection algebras leads to various new results in conventional "commutative" algebraic geometry, which have been inaccessible by purely "commutative" methods. The proposal opens up a wide variety of independent directions for research. One of the basic goals of the proposal is in applying the methods and results developed in one branch of mathematics to a totally different area of mathematics. For example the proposal involves systematic applications of various algebraic techniques to problems in geometry and representation theory. Thus, the proposal enhances the idea of "unity" of mathematics. This proposal also has far reaching applications not only in various areas of mathematics, but also has applications in modern mathematical and theoretical physics, especially in quantum physics. Mathematical science as a whole makes a profound contribution to society. The current proposal goes a long way toward promoting interactions between different disciplines and enhancing scientific understanding.