above, the renormalized character is the scalar-valued character given by the evaluation of φ+ at z = 0 (whereas the evaluation of φ at z = 0 does not necessarily make sense).
Here, we consider the situation where the algebra
with
THEOREM 1.2.–
1 1) Let ℋ be a connected filtered Hopf algebra. Let be the group of those , such that endowed with the convolution product. Any admits a unique Birkhoff decomposition:
[1.30]
where φ− sends 1 to and Ker ε into , and φ+ sends ℋ into . The maps φ- and φ+ are given on Ker ε by the following recursive formulae:
[1.31]
[1.32]
1 2) If the algebra is commutative and if φ is a character, the components φ- and φ+ occurring in the Birkhoff decomposition of χ are characters as well.
PROOF .– The proof goes along the same lines as the proof of Theorem 4 from Connes and Kreimer (1998): for the first assertion, it is immediate from the definition of π that φ- sends Ker ε into
The proof of assertion 2 goes exactly as in Connes and Kreimer (1998) and relies on the following Rota–Baxter relation in
[1.33]
which is easily verified by decomposing a and b into their
with X = φ(x) – Σ(x) φ-(x′)φ(x″) and Y = φ(y) – Σ(y) φ – (y′)φ(y″). Using the formula:
we get:
hence:
We have to compare this expression with:
These two expressions are easily seen to be equal using the commutativity of the algebra
REMARK 1.2.– Assertion 2 admits a more conceptual proof (see the notes by Ebrahimi-Fard in the present volume), which is based on the following recursive expressions for the components of the Birkhoff decomposition: define the Bogoliubov preparation map as the map b : , recursively given by:
[1.34]
Then, the components of φ in the Birkhoff decomposition read:
[1.35]
The Bogoliubov preparation map also writes in a more concise form:
[1.36]
Plugging equation [1.36] into equation [1.35] and setting α := e − φ, we get the following expression for φ−:
[1.37]
[1.38]
where
[1.39]
[1.40]
with β := φ1 * α = e – φ1, and where is the projection on
1.4. Pre-Lie algebras
Pre-Lie algebras are sometimes called Vinberg algebras, as they appear in the work of Vinberg (1963) under the name “left-symmetric algebras” on the classification of homogeneous
