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J. Varilly

 

 

 

Cyclic Cohomology, Hopf Algebras and Quantum Theory

 

Abstract

We give an overview of Non-commutative Geometry, spectral triples and cyclic cohomology. There is a close relation of cyclic cohomology with the theory of charged fields: for instance, Schwinger terms are cyclic cocycles. Hopf algebras have recently appeared as an organizing principle in perturbative calculations
with Feynman graphs. Moreover, Hopf symmetries can be developed by means of a cyclic theory for Hopf algebras.

Index

1. CICLIC COHOMOLOGY AND FREDHOLM MODULES

a. A brief overview of noncommutative geometry
b. Hochschild and cyclic cohomology of algebras
c. Fredholm modules and Schwinger terms

2. HOPF ALGEBRAS AND NONCOMMUTATIVE SYMMETRIES

a.Introduction to Hopf algebras and group duality
b. Compact quantum groups
c. Hopf actions of differential operators: an example

3. HOPF SYMMETRIES IN GEOMETRY AND PHYSYCS

a. The Connes-Kreimer algebra of rooted trees
b. Hopf algebras of Feynman graphs and renormalization
c. Cyclic cohomology of Hopf algebras

4. QUANTIZATION AND NONCOMMUTATIVE HOMOGENEOUS SPACES

a. Chern characters and noncommutative spheres
b. How Moyal products yield compact quantum groups
c. Isospectral deformations of homogeneous spin geometries

 

 

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