Hopf algebra $\mathcal{K}_n$ and universal Chern classes

TL;DR

This paper introduces a new Hopf algebra variant $ K_n$ that acts on noncommutative leaf spaces, showing its cyclic cohomology aligns with universal classes and connecting these to geometric cocycles.

Contribution

It constructs a novel Hopf algebra $ K_n$ acting on noncommutative leaf spaces and establishes its cyclic cohomology as universal, linking algebraic classes to geometric cocycles.

Findings

Cyclic cohomology of $ K_n$ matches that of $( H_n, GL_n)$

Universal Hopf cyclic classes are realized via geometric cocycles

$ K_n$ acts directly on noncommutative models of leaf spaces

Abstract

We construct a variant $\mathcal{K}_n$ of the Hopf algebra $\mathcal{H}_n$, which acts directly on the noncommutative model for the generic space of leaves rather than on its frame bundle. We prove that the Hopf cyclic cohomology of $\mathcal{K}_n$ is isomorphic to that of the pair $(\mathcal{H}_n, {\mathop{\rm GL}_n})$ and thus consists of the universal Hopf cyclic classes. We then realize these classes in terms of geometric cocycles.