Topological Hopf Algebras and Their cyclic cohomology

Hopf代数（量子群）及其上同调为代数提供了不变量。通常代数是复杂的对象来处理，即使在经典的情况下，主要出现在几何空间的坐标。 Hopf代数的主要任务之一是作用于代数，并定义代数的一部分，我们可以更容易地控制。*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起着簿记的作用。他们还计算了他们的霍普夫代数的上同调， 与形式向量场代数的Gelfand-Fuks上同调。 后来，作者和他的合作者注意到，康纳斯和莫斯科维奇所观察到的只是冰山一角。它被扩大到包括coalgberas赋予对称性的霍普夫代数和系数也被添加到理论。* 在这个提议中，我们将Hopf循环上同调推广到拓扑Hopf代数的水平。这使我们能够解决许多悬而未决的问题，自然提出了代数的情况下。 例如正当理由消灭Godbillon-Vey类的情况下，相应的代数霍普夫代数作用于III型代数相关的一般叶理的欧几里德空间。 作为另一个改进，人们观察到，在拓扑Hopf代数的情况下，经典和非经典系数之间的对应是完美的。 这在代数的情况下是缺失的。*我们还试图解决长期存在的问题，在特征类的叶理：是Gelfand-Fuks上同调的唯一来源的特征类的叶理。 我们观察到，有一个自然的一套系数生产的Godbillon时，他试图回答这个问题，他最后发表的论文。我们尝试计算他未计算的复形的上同调，其次数大于2。将培养两名博士生和一名博士后参与项目。主要合作者是Henri Moscovici、塞尔坎Sutlu和Fereshteh Yazdani。

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.