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功能的交换环替换为一个非共同的函数环，该函数环节在某些希尔伯特空间上被认为是线性算子的代数（“量化”）。 “替换''是指在相同的功能空间上找到具有参数h的非交换性乘法（普朗克常数的抽象）时，当h = 0给出了函数的普通乘法时，经典的限制，经典的限制，h中的术语术语（he quassial-classical limit of poisson and osisson结构），将其定义为独立的典范。几年前，有一个强大的定理说，至少（作为h的扩展）可以从泊松结构中恢复到量化，但很少有泊松结构屈服于明确的非交换性变形，而另一方面，使用了许多年前，这些载体上的属性是在许多年前的属性。因此，我们在投影空间的泊松结构。 Poisson的几何形状已在国际层面上进行了多年，但是所提出的问题似乎与代数几何相互作用并不良好，这是该提议主要关注的。它旨在促进我们对全体形态泊松歧管及其可能的非交通变形之间关系的理解。