Algebra and Applications 2. Группа авторов

and similarly for . We can limit ourselves to the kernel of the counit. Any is primitive, hence is obviously a linear isomorphism. Now, for (for some integer n ≥ 2), we can write, using cocommutativity:

      where the x^(j)s are of coradical filtration degree 1, hence primitive. But, we also have:

      [1.29]

      Hence, the element belongs to . It is a linear combination of products of primitive elements by induction hypothesis, hence so is x. We have thus proven that is generated by , which amounts to the surjectivity of φ.

      Now consider a nonzero element , such that φ(u) = 0, and such that d(u) is minimal. We have already proven d(u) ≥ 2. We now compute:

      By minimality hypothesis on d(u), we then get Σ_(u) u′ ⊗ u″ = 0. Hence, u is primitive, that is, d(u) = 1, a contradiction. Hence, φ is injective. The compatibility with the original filtration or graduation is obvious. □

Now, let ℋ : ∪_{n ≥ 0} ℋⁿ be a connected filtered Hopf algebra and let be a commutative unital algebra. We suppose that the components of the filtration are finite-dimensional. The group defined in the previous section depends functorially on the target algebra : in particular, when the Hopf algebra ℋ itself is commutative, the correspondence is a group scheme. In the graded case with finite-dimensional components, it is possible to reconstruct the Hopf algebra ℋ from the group scheme. We have indeed:

      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 G₁(k).

      1.3.7. Renormalization in connected filtered Hopf algebras

      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: