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。代数通常是复杂的对象，即使在经典代数的情况下，通常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，并定义代数的一部分，我们可以更容易地控制。***Hopf循环上同是由Alain Connes和Henri Moscovici定义的。他们观察到某种霍普夫代数在他们著名的指数公式中起着记账的作用。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上同调进行了识别。后来，作者和他的合作者发现，Connes和Moscovici所观察到的只是冰山一角。将其扩大到包含Hopf代数赋予对称性的代数，并将系数添加到理论中。***本文将Hopf循环上同推广到拓扑Hopf代数的水平。这使我们能够解决在代数情况下自然产生的许多悬而未决的问题。例如，在相应的代数Hopf代数作用于欧几里德空间上与一般叶形相关的III型代数的情况下，证明哥德亿维类的湮灭。另一个改进是，在拓扑霍普夫代数的情况下，经典系数和非经典系数之间的对应关系是完美的。这在代数情况中是缺失的。***我们也试图解决长期存在于叶状特征类中的问题：Gelfand-Fuks上同是否是叶状特征类的唯一来源？我们观察到哥德隆在他最后发表的论文中试图回答这个问题时产生了一组自然的系数。我们试图在大于2的次数上计算他未计算的复合体的上同调。***将培养两名博士生和一名博士后参与项目。主要合作伙伴是Henri Moscovici， Serkan 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.