Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

TL;DR

This paper constructs Hopf algebras associated with primitive Lie pseudogroups, explores their cyclic cohomology, and provides explicit cocycle representatives, advancing the understanding of quantum symmetries in infinite-dimensional settings.

Contribution

It introduces a new Hopf algebra framework for primitive Lie pseudogroups and computes their Hopf cyclic cohomology with explicit cocycle representatives.

Findings

Computed relative Hopf cyclic cohomology for the general pseudogroup.

Found explicit cocycle representatives for universal Chern classes.

Determined all Hopf cyclic cohomology groups for local diffeomorphisms of the line.

Abstract

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the \'etale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.