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Holomorphic Poisson structures

全纯泊松结构

基本信息

批准号：
EP/K033654/1
负责人：
Nigel James Hitchin
金额：
$ 34.67万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK033654%2F1
关键词：
Holomorphic Poisson structures

项目摘要

The idea of quantizing a space is to replace the commutative ring of functions by a non-commutative one which is supposed to be realized ("quantized") as an algebra of linear operators on some Hilbert space. "Replacement'' means finding a non-commutative multiplication on the same space of functions with a parameter h (an abstraction of Planck's constant) which when h=0 gives the ordinary multiplication of functions, the classical limit. The term to first order in h (the "quasi-classical" limit) defines a mathematical structure called a Poisson structure. It can be defined independently in differential geometric terms and in fact Kontsevich over 10 years ago proved a powerful theorem which said that at least formally (as an expansion in h) one could go from the Poisson structure back to a quantization, yet very few Poisson structures have yielded to explicit noncommutative deformations. On the other hand non-commutative algebra structures on vector spaces were defined many years ago by the theoretical physicist Sklyanin using elliptic functions, and these induce holomorphic Poisson structures on projective space. We thus have a general principle relating non-commutative geometry and Poisson geometry, but few examples and little understanding of how wide or narrow is the world of Poisson manifolds which admit explicit quantizations. Poisson geometry has been pursued for many years at an international level, but the questions that were posed seemed not to interact well with algebraic geometry, which is what this proposal is mainly concerned with. It is intended to advance our understanding of the relationship between holomorphic Poisson manifolds and their possible non-commutative deformations.

量子化空间的思想是用一个非交换的函数环代替交换的函数环，这个非交换的函数环应该被实现（“量子化”）为某个希尔伯特空间上的线性算子代数。“替换”意味着在同一个函数空间上找到一个非交换乘法，参数为h（普朗克常数的抽象），当h=0时，给出函数的普通乘法，经典极限。h中的一阶项（“准经典”极限）定义了一个称为泊松结构的数学结构。它可以独立地定义在微分几何方面，事实上Kontsevich超过10年前证明了一个强大的定理说，至少在形式上（作为一个扩大h）可以从泊松结构回到量化，但很少泊松结构产生了明确的非交换变形。另一方面，向量空间上的非交换代数结构在许多年前由理论物理学家Sklyanin使用椭圆函数定义，并且这些诱导了射影空间上的全纯泊松结构。因此，我们有一个一般原则有关的非交换几何和泊松几何，但很少有例子和了解如何宽或窄的世界泊松流形承认明确的量化。泊松几何已经追求多年在国际一级，但所提出的问题似乎没有互动以及与代数几何，这是什么这个建议是主要关注。它的目的是推进我们的理解全纯泊松流形和它们可能的非交换变形之间的关系。