Hopf algebras in noncommutative geometry

TL;DR

This paper surveys the application of Hopf algebras in noncommutative geometry, highlighting their roles in index formulas, renormalization, characteristic classes, and quantum groups, with key examples like the Hopf algebra of rooted trees.

Contribution

It provides an introductory overview connecting Hopf algebras to various noncommutative geometric structures and problems, emphasizing their unifying role.

Findings

Relation of rooted trees Hopf algebra to index formulas

Hopf algebra structures in renormalization processes

Construction of noncommutative spherical manifolds as quantum homogeneous spaces

Abstract

We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups.