Hopf algebras, from basics to applications to renormalization

These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a self-contained introduction to the Hopf-algebraic techniques which appear in the work of A. Connes and D. Kreimer on renormalization in Quantum Field Theory [CK1], [CK2], [BF]... Our point of view consists in revisiting a substantial part of their work in the abstract framework of connected graded Hopf algebras, i.e. Hopf algebras endowed with a compatible Z+ grading such that the degree zero component is one-dimensional. Chapter I contains a few elements of Hopf algebra theory which can be found in any good introductory text on the subject ([Ab], [Sw], [Ka]...), as well as some basic tools from algebra which are necessary to understand the coradical filtration of a Hopf algebra. Chapter II deals with connected graded and connected filtered Hopf algebras with emphasis on the convolution product. The main interest of these objects resides in the possibility to implement induction techniques with respect to the grading or the filtration : the starting point is the particular form of the coproduct on a connected filtered Hopf algebra H :