Topological Hopf Algebras and Their cyclic cohomology

拓扑 Hopf 代数及其循环上同调

• 批准号: RGPIN-2018-04039
• 负责人: Rangipour, Bahram
• 金额: $ 1.46万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712723
• 关键词: Topological; Hopf; Algebras; Their; cyclic

Hopf algebras (quantum groups) and their cohomology provides invariants for algebras. Usually algebras are complicated objects to deal with even in the case of classical ones that mostly appear as the coordinates of geometric spaces. One of the main duties of Hopf algebras is to act on algebras and define a part of algebra at which we can control easier. Hopf cyclic cohomology was defined by Alain Connes and Henri Moscovici. They observed that a certain Hopf algebra plays a bookkeeping role in their celebrated index formula. They also calculated the cohomology of their Hopf algebra and identified it with the Gelfand-Fuks cohomology of the algebra of formal vector fields. Later on it was observed by the author and his collaborators that what Connes and Moscovici observed is the tip of an iceberg. It was enlarged to encompass coalgberas endowed with symmetry from Hopf algebras and also the coefficients was added to the theory. In this proposal we extend Hopf cyclic cohomology to the level of topological Hopf algebras. This allows us to solve many of the open questions that has raised naturally in the algebraic cases. For instance justification to the annihilation of Godbillon-Vey classes in case of the corresponding algebraic Hopf algebra act on the type III algebra associated to general foliations on the Euclidean space. As another improvement one observes that in the case of topological Hopf algebras the correspondence between classical and nonclassical coefficients are perfect. This was missing in the algebraic case. We also try to solve the long standing problem in characteristic classes of foliation: is the Gelfand-Fuks cohomology are the only source of characteristic classes of foliations. We observe that there is a natural set of coefficients produced by Godbillon when he tried to answer the question on his very last published paper. We try to compute the cohomology of the complex he left uncalculated at the degree of greater than 2. Two PhD students and one postdoctoral fellow will be trained to involve in project. The main collaborates are Henri Moscovici, Serkan Sutlu, and Fereshteh Yazdani.

Hopf代数(量子群)及其上同调为代数提供了不变量。通常情况下，代数是要处理的复杂对象，甚至在经典代数的情况下也是如此，它们大多表现为几何空间的坐标。Hopf代数的主要职责之一是作用于代数并定义我们可以更容易控制的代数的一部分。 Hopf循环上同调是由Alain Connes和Henri Moscovici定义的。他们观察到，某个霍普夫代数在他们著名的指数公式中扮演着簿记角色。他们还计算了他们的Hopf代数的上同调，并用形式向量场代数的Gelfand-Fuks上同调来确定它。后来，作者和他的合作者观察到，康尼斯和莫斯科维奇观察到的只是冰山一角。它被扩展到包含来自Hopf代数的具有对称性的coalgberas，并将系数添加到该理论中。 在这个方案中，我们将Hopf循环上同调推广到拓扑Hopf代数的层次。这使我们能够解决在代数情况下自然提出的许多未决问题。例如，在相应的代数Hopf代数作用于与欧几里德空间上的一般叶相关联的类型III代数的情况下，证明Godbillon-Vey类的湮灭是正当的。作为另一种改进，人们观察到，在拓扑Hopf代数的情况下，经典系数和非经典系数之间的对应是完美的。在代数案例中，这一点是缺失的。 我们还试图解决叶理特征类中长期存在的问题：Gelfand-Fuks上同调是叶理特征类的唯一来源吗？我们观察到，当戈德比伦试图回答他最后一篇发表的论文中的问题时，他产生了一组自然的系数。我们试着计算他在大于2的次数下未计算的复数的上同调。 两名博士生和一名博士后将接受参与该项目的培训。主要的合作者是Henri Moscovici、Serkan Sutlu和Fereshteh Yazdani。