Symplectic Reflection Algebras

摘要：辛反射代数本课题的主题是将非交换几何的一般机制应用于代数和几何的具体问题。这个提议在看似无关的数学分支之间建立了几个意想不到的新联系，比如非交换代数、代数几何、表示理论和物理学。具体地说，给定一个有限维辛向量空间和该向量空间的有限群辛自同构，研究了相应轨道折的辛分辨和辛变形（交换和非交换）。通过Hochschild上同调，给出了求解各种“弦”拓扑不变量的新方法。因此，该建议的主要思想之一是通过非交换几何技术来接近轨道的代数几何。“辛反射代数”为实现这一思想提供了一个基本工具。辛反射代数的系统使用导致了传统“交换”代数几何中各种新的结果，这些结果是纯“交换”方法无法获得的。这一建议开辟了多种独立的研究方向。该提案的基本目标之一是将在一个数学分支中发展起来的方法和结果应用于完全不同的数学领域。例如，该提案涉及系统地应用各种代数技术来解决几何和表示理论中的问题。从而强化了数学的“统一性”思想。这一建议不仅在数学的各个领域具有深远的应用，而且在现代数学和理论物理，特别是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远的贡献。目前的提议在促进不同学科之间的互动和加强科学理解方面有很大的帮助。

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.