  Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer
approach to renormalization, we augment the latter by a combinatorial, Lie
algebraic point of view. Our results rely both on the properties of the Dynkin
idempotent, one of the fundamental Lie idempotents in the theory of free Lie
algebras, and on properties of Hopf algebras encapsulated in the notion of
associated descent algebras. Besides leading very directly to proofs of the
main combinatorial aspects of the renormalization procedures, the new
techniques give rise to an algebraic approach to the Galois theory of
renormalization. In particular, they do not depend on the geometry underlying
the case of dimensional regularization and the Riemann-Hilbert correspondence.
This is illustrated with a discussion of the BPHZ renormalization scheme.
