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.

量化空间的想法是将函数的交换环替换为非交换环，该非交换环应该被实现（“量化”）为某些希尔伯特空间上的线性算子的代数。 “替换”意味着在具有参数 h（普朗克常数的抽象）的相同函数空间上找到非交换乘法，当 h=0 时给出函数的普通乘法，即经典极限。 h（“准经典”极限）定义了一种称为泊松结构的数学结构，它可以用微分几何术语独立定义，事实上 Kontsevich 十多年前证明了一个强大的定理，该定理至少表明：形式上（作为 h 中的展开）可以从泊松结构返回到量化，但很少有泊松结构产生显式的非交换变形。另一方面，向量空间上的非交换代数结构多年前就被定义了。理论物理学家 Sklyanin 使用椭圆函数，并且这些函数在射影空间上导出全纯泊松结构，因此我们有一个关于非交换几何和泊松几何的一般原理。对于允许显式量化的泊松流形的世界有多宽或多窄，例子很少，也很少了解。泊松几何在国际上已经研究了很多年，但提出的问题似乎与代数几何没有很好的相互作用，而代数几何正是本提案主要关注的问题。它的目的是增进我们对全纯泊松流形及其可能的非交换变形之间关系的理解。