Algebra and Applications 2

PROPOSITION 1.12.–

where is the Lie algebra of infinitesimal characters with values in the base field k, where stands for its enveloping algebra, and (—)° stands for the graded dual .

In the case when the Hopf algebra ℋ is not commutative, it is no longer possible to reconstruct it from G 1( k ).

In this section we describe the renormalization à la Connes-Kreimer (Connes and Kreimer 1998; Kreimer 2002) in the abstract context of connected filtered Hopf algebras: the objects to be renormalized are characters with values in a commutative unital target algebra endowed with a renormalization scheme , that is, a splitting into two subalgebras. An important example is given by the minimal subtraction scheme of the algebra of meromorphic functions of one variable z , where is the algebra of meromorphic functions which are holomorphic at z = 0, and stands for the “polar parts”. Any -valued character φ admits a unique Birkhoff decomposition :

where φ +is an -valued character, and φ (Ker ε ) ⊂ . In the MS scheme case described 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 admits a renormalization scheme , that is, a splitting into two subalgebras:

with . Let be the projection on parallel to .

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 , and that φ +sends Ker ε into . It only remains to check equality φ += φ -* φ , which is an easy computation:

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 -parts. Let φ be a character of ℋ with values in . Suppose that we have φ -( xy ) = φ -( x ) φ − ( y ) for any x, y ∈ ℋ, such that | x | + | y | ≤ d – 1, and compute for x , y , such that | x | + | y | = d :

with X = φ ( x ) – Σ (x) φ -( x ′) φ ( x ″) and Y = φ ( y ) – Σ (y) φ – ( y ′) φ ( y ″). Using the formula: