Hopf algebras and universal Chern classes

Abstract

We construct a variant K n of the Hopf algebra H n , which acts directly on the noncommutative model for the space of leaves of a general foliation rather than on its frame bundle. We prove that the Hopf cyclic cohomology of K n is isomorphic to that of the pair (H n , gl n ) and thus consists of the universal Hopf cyclic Chern classes. We also realize these classes in terms of geometric cocycles.

Noncommutative Geometry, the spectral standpoint

“…In fact the explicit computations involved in the local index formula for this spectral triple dictated two essential new ingredients: first a Hopf algebra H n which governs transverse geometry in codimension n, second a general construction of the cyclic cohomology of Hopf algebras as a far reaching generalization of Lie algebra cohomology [65][66][67] and [1,147,148]. This theory has been vigorously developed by H. Moscovici and B. Rangipour [217][218][219][220], the DGA version of the Hopf cyclic cohomology and the characteristic map from Hopf cyclic to cyclic were investigated by A. Gorokhovsky in [136].…”