Topological Hopf Algebras and Their cyclic cohomology

拓扑 Hopf 代数及其循环上同调

基本信息

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：
2022
资助国家：
加拿大
起止时间：
2022-01-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758886
关键词：
Topological Hopf Algebras Their cyclic

项目摘要

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，代数是需要处理的复杂对象，即使在大多数以几何空间坐标出现的经典代数的情况下也是如此。霍普夫代数的主要职责之一是作用于代数并定义代数中我们可以更容易控制的部分。 Hopf 循环上同调由 Alain Connes 和 Henri Moscovici 定义。他们观察到特定的霍普夫代数在他们著名的指数公式中扮演着记账的角色。他们还计算了 Hopf 代数的上同调，并将其与形式向量场代数的 Gelfand-Fuks 上同调等同起来。后来作者和他的合作者观察到，康尼斯和莫斯科维奇所观察到的只是冰山一角。它被扩大到包含具有霍普夫代数对称性的煤，并且系数也被添加到理论中。在这个提案中，我们将 Hopf 循环上同调扩展到拓扑 Hopf 代数的水平。这使我们能够解决代数案例中自然提出的许多开放性问题。例如，在相应的代数 Hopf 代数作用于与欧几里得空间上的一般叶状结构相关的 III 型代数的情况下，Godbillon-Vey 类的湮灭的论证。作为另一项改进，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关系是完美的。这在代数案例中是缺失的。我们还尝试解决叶状特征类中长期存在的问题：Gelfand-Fuks 上同调是否是叶状特征类的唯一来源。我们观察到，当戈比伦试图回答他最后发表的论文中的问题时，他产生了一组自然的系数。我们尝试计算他未计算的复数的上同调，其次数大于2。将培训两名博士生和一名博士后参与该项目。主要合作者有 Henri Moscovici、Serkan Sutlu 和 Fereshteh Yazdani。