  The Hopf algebra of renormalization in quantum field theory is described at a
general level. The products of fields at a point are assumed to form a
bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of
B, with the structure of a noncommutative bialgebra. When the bialgebra B is
commutative, renormalization turns S(S(B)^+), the double symmetric algebra of
B, into a commutative bialgebra. The usual Hopf algebra of renormalization is
recovered when the elements of B are not renormalised, i.e. when Feynman
diagrams containing one single vertex are not renormalised. When B is the Hopf
algebra of a commutative group, a homomorphism is established between the
bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series.
The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is
given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as
the standard renormalisation procedure for the scalar field.
