Lie algebroids and groupoids, supermanifolds, and noncommutative geometry

李代数胚和群胚、超流形和非交换几何

• 批准号: 0707137
• 负责人: Alan Weinstein
• 金额: $ 28.67万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707137&HistoricalAwards=false
• 关键词: Lie; algebroids; groupoids; supermanifolds; noncommutative

This project will investigate group structures on differentiable stacks and in bimodule categories of algebras, concentrating on the example of Poisson and quantum tori. The project will also study complex Lie algebroids to gain insight into the relative index problem for Cauchy-Riemann structures on contact manifolds, and to construct holomorphic objects associated with generalized complex manifolds. Finally, applications of groupoids will be made to general relativity and other areas where groups are not flexible enough to encode the natural symmetries of a system.In classical geometry, the notion of "space" is commonly identified with that of smooth manifold or some other kind of set with additional structure. But the study of global and local quotient objects, sometimes arising from concrete problems in mathematics, physics, and engineering, often leads beyond the world of smooth manifolds, and even beyond the world of sets, to objects known as (differentiable) stacks. In this new world, and the dual world of algebra, where geometric objects may be identified with algebras of functions, stacks become objects in a new category in which the isomorphisms are what is generally known as Morita equivalences. The project is devoted to understanding the corresponding notions of symmetry in these new worlds.

该项目将在可区分的堆栈和双模块类别的代数类别上研究组结构，集中在泊松和量子托里的示例上。 该项目还将研究复杂的代数，以深入了解接触歧管上的Cauchy-Riemann结构的相对索引问题，并构建与广义复杂歧管相关的全体形态对象。 最后，将分组的应用在一般相对论和组不足以编码系统的自然对称性的其他领域中进行应用。在经典的几何形状中，“空间”的概念通常与平滑流形或其他具有其他结构的其他类型的概念一起识别。 但是，对全球和本地商对象的研究有时是由于数学，物理和工程学中的具体问题引起的，通常会超越平滑流形的世界，甚至超越集合世界的世界，而不是被称为（可区分的）堆栈的对象。 在这个新世界和代数的双重世界中，几何学对象可以用函数代数来识别，堆栈成为新类别中的对象，在这个新类别中，同构是通常称为摩里塔等效性的对象。 该项目致力于理解这些新世界中对称性的相应概念。