Hopf algebras in noncommutative geometry
arXiv:hep-th/0109077
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.
Latex2e, 76 pages, lecture notes for the CIMPA Summer School in Villa de Leyva, Colombia, July 2001; minor corrections, 3 references added
References in corpus (13)
- Gravity coupled with matter and foundation of non-commutative geometry
- Hopf Algebras, Renormalization and Noncommutative Geometry
- Hopf algebras, cyclic cohomology and the transverse index theory
- Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
- Metrics on State Spaces
- A Short Survey of Noncommutative Geometry
- Quantum groupoids
- Metrics on states from actions of compact groups
- The Standard Model as a noncommutative geometry: the low energy regime
- On Overlapping Divergences
- Quantum symmetry groups of noncommutative spheres
- Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees
- Bicovariant Calculus on Twisted ISO(N), Quantum Poincare' Group and Quantum Minkowski Space