Set φ*t := exp(t log φ) for t ∈ k. It coincides with the nth convolution power of φ for any integer n. Hence, φ*t is an -valued character of ℋ for any t ∈ k. Indeed, for any x, y ∈ ℋ, the expression φ*t(xy) – φ*t(x)φ*t(y) is polynomial in t and vanishes on all integers, and hence, vanishes identically. Differentiating with respect to t at t = 0, we immediately find that log φ is an infinitesimal character. □
1.3.6. Group schemes and the Cartier-Milnor-Moore-Quillen theorem
THEOREM 1.1 (Cartier, Milnor, Moore, Quillen).– Let be a cocommutative connected filtered Hopf algebra and let be the Lie algebra of its primitive elements, endowed with the filtration induced by the one of, which in turns induces a filtration on the enveloping algebra. Then, andare the isomorphic as filtered Hopf algebras. Ifis graded, then the two Hopf algebras are isomorphic as graded Hopf algebras.
PROOF.– The following proof is borrowed from Foissy’s thesis. The embedding obviously induces an algebra morphism
[1.27]
It is easy to show that φ is also a coalgebra morphism. It remains to show that φ is surjective, injective and respects the filtrations. Let us first prove the surjectivity by induction on the coradical filtration degree:
[1.28]
Set , 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:
Set φ*t := exp(t log φ) for t ∈ k. It coincides with the nth convolution power of φ for any integer n. Hence, φ*t is an 