Hopf algebras,

from basics to applications to renormalization

Revised and updated version, may 2006

Dominique Manchon^{1}

Introduction

These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a self-contained introduction to the Hopf-algebraic techniques which appear in the work of A. Connes and D. Kreimer on renormalization in Quantum Field Theory [CK1], [CK2], [BF]... Our point of view consists in revisiting a substantial part of their work in the abstract framework of connected graded Hopf algebras, i.e. Hopf algebras endowed with a compatible ℤ_{+} grading such that the degree zero component is one-dimensional.

Chapter I contains a few elements of Hopf algebra theory which can be found in any good introductory text on the subject ([Ab], [Sw], [Ka]...), as well as some basic tools from algebra which are necessary to understand the coradical filtration of a Hopf algebra.

Chapter II deals with connected graded and connected filtered Hopf algebras with emphasis on the convolution product. The main interest of these objects resides in the possibility to implement induction techniques with respect to the grading or the filtration : the starting point is the particular form of the coproduct on a connected filtered Hopf algebra H :

Δx = x ⊗ 1 + 1 ⊗ x + terms involving elements of strictly lower filtration degree.

We derive the Birkhoff decomposition of any linear map on H with values into any commutative unital algebra A which sends the unit 1 to the unit 1_{A}, provided the algebra A is endowed with a renormalization scheme, i.e. a splitting :

A = A_{−} ⊕ A_{+}

into two subalgebras, where A_{+} contains the unit 1_{A}. Those maps indeed form a group G under the convolution product, and the Birkhoff decomposition of any φ ∈ G reads :

φ = φ_{−}^{∗−1} ∗ φ_{+},

1 Université Blaise Pascal, CNRS - UMR 6620. manchon@math.univ-bpclermont.fr

where φ_{−} and φ_{+} are elements of G, φ_{−} sends any element of H of positive degree to an element of A_{−}, , and φ_{+} is a map with values in A_{+}. As an example, A is the algebra of germs of meromorphic functions at z_{0} ∈ ℂ endowed with the minimal subtraction scheme, i.e. A_{+} is the algebra of germs of holomorphic functions at z_{0}, and A_{−} is the subalgebra of polar parts (z − z_{0})^{−1}ℂ[(z − z_{0})^{−1}].

The Birkhoff decomposition respects two particular subgroups of G, namely the group G_{1} of algebra morphisms from H into A (characters), and the group G_{2} of elements φ of G such that φ(xy) = φ(yx) (cocycles). We end up this chapter with some examples. The Birkhoff decomposition in this abstract algebraic setting has applications beyond quantum field theory, for example in number theory where it can be used to define multiple zeta values at nonpositive arguments ([GZ], [MP2]).

Chapter III is devoted to Feynman graphs, which are the original example coming from quantum field theory, where D. Kreimer discovered the underlying Hopf algebra structure.

In Chapter IV we define a bijective map R̃ : G → 𝔤 where 𝔤 stands for the linear space of maps on H with values into A which sends the unit 1 to 0. It is uniquely defined by means of the equation :

φ ∘ Y = φ ∗ R̃(φ),

where Y : H → H is the natural biderivation x ↦ |x|.x associated to the graduation. We give an explicit formula for this renormalization map and for its inverse. Finally, along the lines of [CK2] we explore more closely the interplay between the renormalization map R̃ and Birkhoff decomposition. When A is the algebra of germs of meromorphic functions at z_{0} = 0 endowed with the minimal subtraction scheme, the renormalization map R̃ is closely related to the β-function of [CK2] as follows : for any φ ∈ G_{1} with trivial positive part in the Birkhoff decomposition (i.e. φ_{+} = u_{A} ∘ ε, where ε is the co-unit of H and u_{A} is the unit map of A), we have :

β(φ) = zR̃(φ).

(see theorem IV.4.4).

These notes find their origin in a workshop on renormalization in Quantum Field Theory held in Nancy every monday during year 2000-2001. I would like to thank all participants to this workshop, among them specially Philippe Bonneau, Malte Henkel, Mohsen Masmoudi, André Roux, Jérémie Unterberger and Tilmann Wurzbacher. Many thanks as well to the Mathematics Departments of Universidad de Los Andes and Universidad La Nacional at Bogota for the friendly atmosphere of this conference. I would like to thank Luis Fernandez and Sylvie Paycha for valuable remarks on previous versions of this text, as well as Kurusch Ebrahimi-Fard, Dirk Kreimer and Li Guo for illuminating discussions.

Enfin, un grand merci à toute l’équipe des Rencontres Mathématiques de Glanon pour avoir bien voulu publier ces notes dans ce volume des comptes-rendus.

Contents :

Introduction

I. A few elements of Hopf algebra theory

I.1. Tensor algebra I.2. Algebras I.3. Coalgebras I.4. Convolution product I.5. Bialgebras and Hopf algebras I.6. Examples I.7. Some properties of Hopf algebras

II. Convolution product and regularization

II.1. Connected graded bialgebras II.2. Connected filtered bialgebras II.3. The convolution product II.4. Algebra morphisms and cocycles II.5. Birkhoff decomposition II.6. The BCH approach to Birkhoff decomposition II.7. Renormalized traces and characters II.8. More on the graduation II.9. Examples

III. Hopf algebras of Feynman graphs

III.1. Discarding exterior structures III.2. Operations on Feynman graphs III.3. The graded Hopf algebra structure III.4. External structures

IV. An approach to the renormalization group

III.1. The renormalization map III.2. Inverting R̃ : the scattering map III.3. The residue III.4. Renormalization map and Birkhoff decomposition

References

I. A few elements of Hopf algebra theory

I.1. Tensor algebra

Let k be any field, and let A, B be vector spaces over k. The tensor product A ⊗ B is a vector space over k which satisfies the following universal property : there exists a bilinear map :

\begin{matrix}ι : A × B & ⟶ A ⊗ B \\ (a, b) & ⟼ a ⊗ b \end{matrix}

such that for any k-vector space C and for any bilinear map f from A × B into C there is a unique linear map f̃ : A ⊗ B → C such that f = f̃∘ ι :

\xymatrix{A ⊗ B \ar[dr]^{_{f}} \\ A × B \ar[u]^{ι} \ar[r]^{f} & C}

Proposition I.1.1. Tensor product A ⊗ B exists and is unique up to isomorphism.

Proof. We show uniqueness first : if (T_{1}, ι_{1}) and (T_{2}, ι_{2}) are both candidates for playing the role of the tensor product, then the universal property applied to both tells us that there exist ϕ : T_{1} → T_{2} and ψ : T_{2} → T_{1} such that ι_{2} = ϕ ∘ ι_{1} and ι_{1} = ψ ∘ ι_{2}.

\xymatrix{T_{1} \ar[dr]^{ϕ} \\ A × B \ar[u]^{ι_{1}} \ar[r]_{ι_{2}} & T_{2} \ar[ul]^{ψ}}

Applying the universal property twice again shows that ψ ∘ ϕ = I_{T_{1}} and ϕ ∘ ψ = I_{T_{2}}, whence uniqueness of the tensor product up to linear isomorphism.

For the existence we shall need axiom of choice in the infinite-dimensional case : let (e_{i})_{i∈E} and (f_{j})_{j∈F} be bases of A and B respectively. Then we take as A ⊗ B the vector space freely generated by (c_{ij})_{i∈E,j∈F}, and we set ι(e_{i}, f_{j}) = e_{i} ⊗f_{j} = c_{ij}. For any bilinear map ϕ from A × B to another vector space C it is clear that the linear map ϕ̃ from A ⊗ B to C defined by :

ϕ̃(c_{ij}) = ϕ(e_{i}, f_{j})

is the only one such that ϕ̃∘ ι = ϕ. The vector space A ⊗ B constructed above fulfills then the universal property.

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Elements a ⊗ b in A ⊗ B are called indecomposable. They clearly generate A ⊗ B. Tensor products k ⊗ A and A ⊗ k are canonically identified with A via 1 ⊗ a ≃ a ≃ a ⊗ 1. When three vector spaces A, B, C are involved there is an isomorphism :

\begin{matrix}α : (A ⊗ B) ⊗ C & ⟶ A ⊗ (B ⊗ C) \\ (a ⊗ b) ⊗ c & ⟼ a ⊗ (b ⊗ c). \end{matrix}

Note that this isomorphism is not canonical, because tensor product A ⊗ B itself is only defined up to isomorphism. We shall denote by A ⊗ B ⊗ C any of these two versions of the iterated tensor product.

Let A and B be two vector spaces. The flip τ : A⊗B → B⊗A defined by τ(a⊗b) = b⊗a is an isomorphism. This generalizes to any finite collection of vector spaces as follows : any permutation σ ∈ S_{n} defines a map :

\begin{matrix}τ_{σ} : A_{1} ⊗ ⋯ ⊗ A_{n} & ⟶ A_{σ^{−1}1} ⊗ ⋯ ⊗ A_{σ^{−1}n} \\ a_{1} ⊗ ⋯ ⊗ a_{n} & ⟼ a_{σ^{−1}1} ⊗ ⋯ ⊗ a_{σ^{−1}n}. \end{matrix}

In particular the group S_{n} acts on the tensor power A^{⊗n} by automorphisms. It makes then sense to consider the tensor power A^{⊗F} for any finite set, by choosing any arbitrary total order on F.

Now let us consider four k-vector spaces A_{1}, A_{2}, B_{1}, B_{2}.

Proposition I.1.2. There is a canonical injection :

L(A_{1}, A_{2}) ⊗ L(B_{1}, B_{2}) ⟶_{∼}^{j} L(A_{1} ⊗ B_{1}, A_{2} ⊗ B_{2})

given by :

(j(f ⊗ g))(a ⊗ b) = f(a) ⊗ g(b).

In the case when all the spaces are finite-dimensional, this injection is an isomorphism.

Proof. The space L(A_{1} ⊗ B_{1}, A_{2} ⊗ B_{2}) together with the bilinear map :

\begin{matrix}ι : L(A_{1}, A_{2}) × L(B_{1}, B_{2}) & ⟶ L(A_{1} ⊗ B_{1}, A_{2} ⊗ B_{2}) \\ (f, g) & ⟼ (a ⊗ b ↦ f(a) ⊗ g(b)) \end{matrix}

yields the map j, which is easily seen to be injective. In the finite-dimensional case we can either compute the dimensions or verify that the space L(A_{1} ⊗ B_{1}, A_{2} ⊗ B_{2}) together with the bilinear map ι fulfills the universal property. The details are left to the reader.

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Remark : There are “super” versions of the tensor product for super-vector spaces (i.e. ℤ_{2}-graded vector spaces). Elements of vector spaces and linear morphisms are decomposed into degree zero (even) and degree one (odd) components. Then proposition I.1.2 must be modified by a minus sign if both g and a are odd : this is the Koszul rule of signs :

(f ⊗ g)(a ⊗ b) = (−1)^{|g||a|}f(a) ⊗ g(b),

according to which any transposition in such a formula brings a sign which is minus if and only if the two elements involved in the transposition are odd. We shall not use this ℤ_{2}-graded framework in the sequel.

I.2. Algebras and modules

I.2.1. Basic definitions

A k-algebra is by definition a k-vector space A together with a bilinear map m : A⊗A → A which is associative. The associativity is expressed by the commutativity of the following diagram :

\xymatrix{A ⊗ A ⊗ A \ar[d]^{I⊗m} \ar[r]^{m⊗I} & A ⊗ A \ar[d]^{m} \\ A ⊗ A \ar[r]^{m} & A}

The algebra A is unital if moreover there is a unit 1 in it. This is expressed by the commutativity of the following diagram :

\xymatrix{k ⊗ A \ar[r]^{u⊗I} \ar[dr]^{∼} & A ⊗ A \ar[d]_{m} & A ⊗ k \ar[l]_{I⊗u} \ar[dl]_{∼} \\  & A}

where u is the map from k to A defined by u(λ) = λ1. The algebra A is commutative if m ∘ τ = m, where τ : A ⊗ A → A ⊗ A is the flip, defined by τ(a ⊗ b) = b ⊗ a.

A subspace J ⊂ A is called a subalgebra (resp. a left ideal, right ideal, two-sided ideal) of A if m(J ⊗ J) (resp. m(J ⊗ A), m(A ⊗ J), m(J ⊗ A + A ⊗ J) is included in J.

I.2.2. Algebras and tensor product

To any vector space V we can associate its tensor algebra T (V). As a vector space it is defined by :

T (V) = ⨁_{k≥0} V^{⊗k},

with V^{⊗0} = k and V^{⊗k+1} := V ⊗ V^{⊗k}. The product is given by the concatenation :

m(v_{1} ⊗ ⋯ ⊗ v_{p}, v_{p+1} ⊗ ⋯ ⊗ v_{p+q}) = v_{1} ⊗ ⋯ ⊗ v_{p+q}.

The embedding of k = V^{⊗0} into T (V) gives the unit map u. Tensor algebra T (V) is also called the free (unital) algebra generated by V. This algebra is characterized by the following universal property : for any linear map φ from V to an algebra A there is a unique algebra morphism φ̃ from T (V) to A extending φ. The fact that this property characterizes T (V) up to isomorphism is an easy exercise left to the reader.

Let A and B be unital k-algebras. We put a unital algebra structure on A ⊗ B in the following way :

(a_{1} ⊗ b_{1}).(a_{2} ⊗ b_{2}) = a_{1}a_{2} ⊗ b_{1}b_{2}.

The unit element 1_{A⊗B} is given by 1_{A} ⊗ 1_{B}, and the associativity is clear. This multiplication is given by :

m_{A⊗B} = (m_{A} ⊗ m_{B}) ∘ τ_{23},

where τ_{23} : A ⊗ B ⊗ A ⊗ B → A ⊗ A ⊗ B ⊗ B is defined by the flip of the two middle factors :

τ_{23}(a_{1} ⊗ b_{1} ⊗ a_{2} ⊗ b_{2}) = a_{1} ⊗ a_{2} ⊗ b_{1} ⊗ b_{2}.

I.2.3. Modules

Let A be any unital algebra. A left A-module is a k-vector space M together with a map α : A ⊗ M → M such that the following diagrams commute :

\xymatrix{A ⊗ A ⊗ M \ar[d]^{I⊗α} \ar[r]^{m⊗I} & A ⊗ M \ar[d]^{α} & k ⊗ M \ar[r]^{u⊗I} \ar[dr]^{∼} & A ⊗ M \ar[d]^{α} \\ A ⊗ M \ar[r]^{α} & M &  & A}

The map α is called the action of the algebra A on M. For any a ∈ A and m ∈ M we usually denote by a.m the action α(a ⊗ m) of a on m. The two diagrams above express the identities :

(a.b).m = a.(b.m), 1.m = m

for any a, b ∈ A and m ∈ M. The right A-module are defined similarly, replacing A ⊗ M with M ⊗ A (details are left to the reader). A linear subspace N of a left A-module M is called a submodule if α(A ⊗ N) ⊂ N. The intersection of all left submodules of M containing a subset P is called the left submodule generated by P.

A left module M is simple if it does not contain any submodule different from {0} or M itself. If a left module M can be written as a direct sum of simple modules, M is said to be semi-simple.

Proposition I.2.1. For any left maximal ideal J of an algebra A the quotient A/J is a simple left A-module, and conversely any simple left A-module is isomorphic to a simple left A-module of this form.

Proof. The first assertion is immediate. Conversely, let M a simple left module and let m ∈ M − {0}. Let J_{m} be the annihilator of m. It is a left ideal of A, and by simplicity of M the map :

\begin{matrix}ϕ_{m} : A & ⟶ M \\ a & ⟼ a.m \end{matrix}

gives rise to an morphism of left A-modules from A/J_{m} onto M. It is injective by definition of J_{m}, and surjectivity comes from the simplicity of the module M. So ϕ_{m} is an isomorphism. The left ideal J_{m} is maximal, which proves the proposition.

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Now let M be an A-module. We denote by A′_{M} the algebra of endomorphisms of M as an A-module, and we denote by A′′_{M} the algebra of endomorphisms of M as an A′_{M}-module. Clearly any a ∈ A gives rise to an element of A′′_{M}. Following N. Jacobson [J Chap. 4.3] we shall give a proof of an important density theorem :

Theorem I.2.2. Let M be a semi-simple A-module, and let x_{1}, …, x_{n} a finite collection of elements of M. Then for any a′′ ∈ A′′_{M} there exists an element a ∈ A such that ax_{i} = a′′x_{i} for any i = 1, …, n.

Proof. First notice that any A-submodule N of M is an A′′_{M}-submodule. To see this write (thanks to semi-simplicity) M = N ⊕ T where T is another A-submodule of M. The projection e on N with respect to this decomposition is an element of A′_{M}. For any a′′ ∈ A′′_{M} we have then :

a′′(N) = a′′ ∘ e(M) = e ∘ a′′(M) ⊂ N.

Consider then for any fixed positive integer n the semi-simple module M^{n}, direct sum of n copies of M. The algebra A′_{M^{n}} coincides with the algebra of n × n matrices over A′_{M}, and the diagonal matrices over A′′_{M} form a subalgebra of A′′_{M^{n}}, and thus realize an embedding of A′′_{M} into A′′_{M^{n}}.

Consider x = (x_{1}, …, x_{n}) ∈ M^{n}. Then N = A.x is an A-submodule of M^{n}. So it is an A′′_{M^{n}}-submodule, hence an A′′_{M}-submodule via the diagonal embedding above. Then for any a′′ ∈ A′′_{M} there exists a ∈ A such that a′′x = ax, which proves the theorem.

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Corollary I.2.3. On a semi-simple finite-dimensional module M the natural map from A into A′′_{M} is surjective.

I.2.4. The Jacobson radical

Let A be a k-algebra. The radical rad M of a left module is by definition the intersection of all maximal submodules of M. When the module M is the algebra A itself, the radical rad A is the intersection of all maximal left ideals, and is called the Jacobson radical of the algebra A.

We shall give an alternative definition of the Jacobson radical : a primitive ideal is the annihilator of a simple module. In view of proposition I.2.1, any primitive ideal is the annihilator of A/J where J is a maximal left ideal. Of course a primitive ideal is two-sided.

Lemma I.2.4. Any primitive ideal is an intersection of maximal left ideals.

Proof. Any primitive ideal J is by definition the annihilator of a simple module M. The annihilator J_{m} of any m ∈ M − {0} is then a maximal left ideal containing J, and it is clear that we have :

J = ⋂_{m∈M−{0}} J_{m}.

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Proposition I.2.5. The Jacobson radical of A is the intersection of its primitive ideals.

Proof. Let us call P the intersection of all primitive ideals of A. By lemma I.2.4 and proposition I.2.1, P is indeed the intersection of all maximal left ideals.

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Lemma I.2.6 (Nakayama’s lemma). Let M a finitely generated A-module, and let N, L two submodules of M such that M = L + N and N ⊂ rad M. Then L = M.

Proof. Suppose that L is strictly contained in M. As M is finitely generated there exists a maximal nontrivial submodule L̃ containing L. It contains N as well by definition of rad M. Then L̃ contains L + N, so L + N cannot be equal to M.

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Corollary I.2.7. The Jacobson radical of a finite-dimensional algebra is nilpotent.

Proof. Let A a finite-dimensional algebra with radical R. Observe first that for any A-module M we have the inclusion :

R.M ⊂ rad M.

Indeed any maximal submodule N of M contains J.M where J is a primitive ideal, namely the annihilator of M/N. Hence any maximal submodule of M contains R.M. Suppose now that A is finite dimensional and that J is an ideal of A such that R.J = J. A fortiori rad J = J. Applying Nakayama’s lemma I.2.6 to M = J and L = {0} we get J = {0}. We immediately deduce from this fact that for any positive integer n, R^{n} either contains strictly R^{n+1} or is equal to {0}. As A is finite-dimensional R^{n} is indeed equal to {0} for some n.

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Remark : although the definition of the Jacobson ideal is not symmetric (because we used left ideals and left modules), the Jacobson ideal itself is a symmetric notion : in other words the Jacobson radicals of algebras A and A^{opp} coincide. In order to see this one can show that Rad A is the biggest two-sided ideal J such that 1 − x is invertible for any x ∈ J [B §6.3]. This definition is indeed symmetric.

I.2.5. Maximal two-sided ideals

It is easily seen that any maximal two-sided ideal in an algebra A is primitive. The converse is false in general : for example, in the enveloping algebra of the non-trivial twodimensional Lie algebra, the ideal {0} is primitive but not maximal [Di]. However we have the following result :

Proposition I.2.8. Any finite-codimensional primitive ideal is maximal.

Before giving a proof of this result, we need the following definitions : an algebra A is simple if it does not contain any proper two-sided ideal. A division algebra is an algebra such that any nonzero element is inversible.

Lemma I.2.9. Let D be a division algebra. Then the algebra M_{n}(D) of square n × n matrices over D is simple.

Proof. Let us consider for any i, j ∈ {1, …, n} the elementary matrix e_{ij} with vanishing entries except the one on the i^{th} row and j^{th} column which is equal to the unit of D. Denote by I_{i} the left ideal of M_{n}(D) consisting of matrices such that all columns vanish except the i^{th} column. These left ideals are all simple and isomorphic as M_{n}(D)-modules. Now let I a nonzero two-sided ideal of M_{n}(D). Let X a nonzero element of I, and x_{kl} a nonzero entry of the matrix X. Then for any i ∈ {0, …, n} the product e_{ik}X belongs to I_{i} ∩ I and is different from 0.

Then I_{i} ∩ I ≠ {0}. As I_{i} is simple as a left module for each i that means that I contains all the left ideals I_{i}, and hence I = M_{n}(D).

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There is a converse to this result, namely any simple artinian algebra (a fortiori any simple finite-dimensional algebra) is isomorphic to M_{n}(D) where D is a division algebra. This is a particular case of the Wedderburn-Artin theorem, which gives a complete description of semi-simple algebras [DF Chap. 1], [DK Chap. 2].

Proof of Proposition I.2.8 : a finite-dimensional primitive ideal I is the annihilator of a simple finite-dimensional module M. By simplicity of M the algebra D = A′_{M} is a division algebra (Schur’s lemma). The action of A on M yields (thanks to corollary I.2.3) a surjective algebra morphism from A onto A′′_{M}, and hence an algebra isomorphism from A/J onto A′′_{M}. But A′′_{M} is a matrix algebra over D, and then is simple (according to lemma I.2.7). So A/J is a simple algebra, which amounts to say that J is maximal as a two-sided ideal.

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Corollary I.2.10. In a finite-dimensional algebra, primitive ideals and maximal two-sided ideals coincide. In particular the Jacobson radical is the intersection of all maximal two-sided ideals in this case.

I.3. Coalgebras and comodules

This paragraph is mostly borrowed from M.E. Sweedler’s book [Sw], particularly chapters 1, 2, 8 and 9.

I.3.1. Coalgebras

Coalgebras are objects wich are somehow dual to algebras : axioms for coalgebras are derived from axioms for algebras by reversing the arrows of the corresponding diagrams :

A k-coalgebra is by definition a k-vector space C together with a bilinear map Δ : C → C ⊗C which is co-associative. The co-associativity is expressed by the commutativity of the following diagram :

\xymatrix{C ⊗ C ⊗ C & C ⊗ C \ar[l]_{Δ⊗I} \\ C ⊗ C \ar[u]_{I⊗Δ} & C \ar[l]_{Δ} \ar[u]_{Δ}}

Coalgebra C is co-unital if moreover there is a co-unit ε such that the following diagram commutes :

\xymatrix{k ⊗ C & C ⊗ C \ar[l]_{ε⊗I} \ar[r]^{I⊗ε} & C ⊗ k \\  & C \ar[u]^{Δ} \ar[ul]_{∼} \ar[ur]^{∼}}

A subspace J ⊂ C is called a subcoalgebra (resp. a left coideal, right coideal, two-sided coideal) of C if Δ(J) is contained in J ⊗ J (resp. C ⊗ J, J ⊗ C, J ⊗ C + C ⊗ J) is included in J. The duality alluded to above can be made more precise :

Proposition I.3.1. 1)The linear dual C^{∗} of a co-unital coalgebra C is a unital algebra, with product (resp. unit map) the transpose of the coproduct (resp. of the co-unity). 2)Let J a linear subspace of C. Denote by J^{⊥} the orthogonal of J in C^{∗}. Then :

J is a two-sided coideal if and only if J^{⊥} is a subalgebra of C^{∗}.

J is a left coideal if and only if J^{⊥} is a left ideal of C^{∗}.

J is a right coideal if and only if J^{⊥} is a right ideal of C^{∗}.

J is a subcoalgebra if and only if J^{⊥} is a two-sided ideal of C^{∗}.

Proof. For any subspace K of C^{∗} we shall denote by K^{⊥} the subspace of those elements of C on which any element of K vanishes. It coincides with the intersection of the orthogonal of K with C, via the canonical embedding C ⟶ C^{∗∗}. So we have for any linear subspaces J ⊂ C and K ⊂ C^{∗} :

J^{⊥⊥} = J, K^{⊥⊥} ⊃ K.

Suppose that J is a two-sided coideal. Take any ξ, η in J^{⊥}. For any x ∈ J we have :

< ξη, x >=< ξ ⊗ η, Δx >= 0,

as Δx ⊂ J ⊗ C + C ⊗ J. So J^{⊥} is a subalgebra of C^{∗}. Conversely if J^{⊥} is a subalgebra then :

ΔJ ⊂ (J^{⊥} ⊗ J^{⊥})^{⊥} = J ⊗ C + C ⊗ J,

which proves the first assertion. We leave it to the reader as an exercice to prove the three other assertions along the same lines. Dually we have the following :

Proposition I.3.2. Let K a linear subspace of C^{∗}. Then :

K^{⊥} is a two-sided coideal if and only if K is a subalgebra of C^{∗}.

K^{⊥} is a left coideal if and only if K is a left ideal of C^{∗}.

K^{⊥} is a right coideal if and only if K is a rightt ideal of C^{∗}.

K^{⊥} is a subcoalgebra if and only if K is a two-sided ideal of C^{∗}.

The linear dual (C ⊗ C)^{∗} naturally contains the tensor product C^{∗} ⊗ C^{∗}. Take as a multiplication the restriction of ^{t}Δ to C^{∗} ⊗ C^{∗} :

m =^{t} Δ : C^{∗} ⊗ C^{∗} ⟶ C^{∗},

and put u =^{t} ε : k → C^{∗}. It is easily seen, by just reverting the arrows of the corresponding diagrams, that coassociativity of Δ implies associativity of m, and that the co-unit property for ε implies that u is a unit.

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The coalgebra C is cocommutative if τ ∘ Δ = Δ, where τ : C ⊗ C → C ⊗ C is the flip. It will be convenient to use Sweedler’s notation :

Δx = ∑_{(x)} x_{1} ⊗ x_{2}.

Cocommutativity expresses then as :

∑_{(x)} x_{1} ⊗ x_{2} = ∑_{(x)} x_{2} ⊗ x_{1}.

Coassociativity reads in Sweedler’s notation :

(Δ ⊗ I) ∘ Δ(x) = ∑_{(x)} x_{1:1} ⊗ x_{1:2} ⊗ x_{2} = ∑_{(x)} x_{1} ⊗ x_{2:1} ⊗ x_{2:2} = (I ⊗ Δ) ∘ Δ(x),

We shall sometimes write the iterated coproduct as :

∑_{(x)} x_{1} ⊗ x_{2} ⊗ x_{3}.

Sometimes we shall even mix the two ways of using Sweedler’s notation for the iterated coproduct, in the case we want to keep partially track of how we have constructed it [DNR]. For example,

Δ_{3}(x) = (I ⊗ Δ ⊗ I) ∘ (Δ ⊗ I) ∘ Δ(x)

= (I ⊗ Δ ⊗ I)(∑_{(x)} x_{1} ⊗ x_{2} ⊗ x_{3})

= ∑_{(x)} x_{1} ⊗ x_{2:1} ⊗ x_{2:2} ⊗ x_{3}.

To any vector space V we can associate its tensor coalgebra T^{c}(V). It is isomorphic to T (V) as a vector space. The coproduct is given by the deconcatenation :

Δ(v_{1} ⊗ ⋯ ⊗ v_{n}) = ∑_{p=0}^{n}(v_{1} ⊗ ⋯ ⊗ v_{p}) ⨂(v_{p+1} ⊗ ⋯ ⊗ v_{n}).

The co-unit is given by the natural projection of T^{c}(V) onto k.

Let C and D be unital k-coalgebras. We put a co-unital coalgebra structure on C ⊗D in the following way : the comultiplication is given by :

Δ_{C⊗D} = τ_{23} ∘ (Δ_{C} ⊗ Δ_{D}),

where τ_{23} is again the flip of the two middle factors, and the co-unity is given by ε_{C⊗D} = ε_{C} ⊗ ε_{D}.

I.3.2. Comodules

Let C be any co-unital coalgebra. A left C-comodule is a k-vector space M together with a map Φ : M → C ⊗ M such that the following diagrams commute :

\xymatrix{C ⊗ C ⊗ M & C ⊗ M \ar[l]_{Δ⊗I} & k ⊗ M & C ⊗ M \ar[l]_{ε⊗I} \\ C ⊗ M \ar[u]_{I⊗Φ} & M \ar[l]_{Φ} \ar[u]_{Φ} &  & C \ar[u]_{Φ} \ar[ul]_{∼}}

The notion of right C-comodule is defined similarly. A linear subspace N of a left Ccomodule M is called a subcomodule if Φ(C) ⊂ C ⊗ N. The intersection of all left subcomodules of M containing a subset P is again a subcomodule, called the left subcomodule generated by P. It will be convenient to use again Sweedler’s notation :

Φ(m) = ∑_{(m)} m_{1} ⊗ m_{0},

with m_{0} ∈ M and m_{1} ∈ C. Comodule property reads in Sweedler’s notation :

(Φ ⊗ I) ∘ Φ(m) = ∑_{(x)} m_{1:1} ⊗ m_{1:2} ⊗ m_{0} = ∑_{(m)} m_{1} ⊗ m_{0:1} ⊗ m_{0:0} = (I ⊗ Δ) ∘ Φ(m),

We shall sometimes write the iterated coproduct as :

∑_{(m)} m_{1} ⊗ m_{2} ⊗ m_{0}.

For a right comodule we have a similar behaviour with m_{0} on the left. The notion of comodule is dual to the notion of module in the sense that any left (resp. right) Ccomodule M admits a right (resp. left) C^{∗}-module structure. To be precise suppose for the moment that Φ is any linear map from M to M ⊗ C, and define α_{Φ} : C^{∗} ⊗ M → M as the following composition :

\xymatrix{C^{∗} ⊗ M \ar[r]^{I⊗Φ} & C^{∗} ⊗ M ⊗ C \ar[r]^{τ⊗I} & M ⊗ C^{∗} ⊗ C \ar[r]^{I⊗<−,−>} & M ⊗ k \ar[r]^{∼} & M.}

Then :

Proposition I.3.3. (M, Φ) is a right (resp. left) C-comodule if and only is (M, α_{Φ}) is a left (resp. right) C^{∗}-module.

Proof. See Sweedler [Sw] section 2.1.

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Note that the duality property is not perfect : if the linear dual of a coalgebra is always an algebra, the linear dual of an algebra is not in general a coalgebra. However the restricted dual A^{∘} of an algebra A is a coalgebra. It is defined as the space of linear forms on A vanishing on some finite-codimensional ideal. Along the same lines, for a coalgebra C the only left C^{∗}-modules related to a right C-comodule structure via proposition I.3.3 are the rational left C^{∗}-modules, i.e. those left modules such that the linear map :

\begin{matrix}ρ : M & ⟶ L(C^{∗}, M) \\ m & ⟼ (x ↦ x.m) \end{matrix}

has image included in M ⊗ C via the embedding :

\begin{matrix}j : M ⊗ C & ⟶ L(C^{∗}, M) \\ m ⊗ c & ⟼ (x ↦< x, c > m). \end{matrix}

See [Sw] for details. We come now to the fundamental theorem of comodule structure theory :

Theorem I.3.4. Let M be a left comodule over a coalgebra C. For any element m ∈ M the subcomodule generated by m is finite-dimensional.

Proof. There is a finite collection (c_{i})_{i=1···s} of linearly independant elements of C and a collection (m_{i})_{i=1···s} of elements of M such that

Φ(m) = ∑_{i=1}^{s} m_{i} ⊗ c_{i}.

Let N be the linear subspace of M generated by m_{1}, …, m_{s}. Let us show that N is a left subcomodule of M. First note that thanks to the co-unit axiom we have :

m = (I ⊗ ε) ∘ Φ(m) = ∑_{j=1}^{s} ε(c_{j})m_{j},

hence m ∈ N. On the other hand, considering linear forms (f_{i})_{i=1···s} on C such that f_{i}(c_{j}) = δ_{i}^{j} we have :

Φ(m_{i}) = (I ⊗ I ⊗ f_{i})(Φ(m_{i}) ⊗ c_{i})

= (I ⊗ I ⊗ f_{i})(∑_{j=1}^{s} Φ(m_{j}) ⊗ c_{j})

= (I ⊗ I ⊗ f_{i}) ∘ (Φ ⊗ I) ∘ Φ(m) = (I ⊗ I ⊗ f_{i}) ∘ (I ⊗ Δ) ∘ Φ(m)

= ∑_{j=1}^{s} m_{j} ⊗ (I ⊗ f_{i})(Δc_{j}),

whence Φ(m_{i}) ∈ N ⊗ C, which proves the theorem.

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Corollary I.3.5. Let M be a left comodule over a coalgebra C. Then any left subcomodule of M generated by a finite set is finite-dimensional.

Proof. remark that if P = {m_{1}, …, m_{n}}, the left subcomodule generated by P is the sum of the left comodules generated by the m_{j}’s, and then apply theorem I.3.4.

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I.3.3. Structure of coalgebras

Let C be a coalgebra. Any intersection of subcoalgebras is a subcoalgebra. To see this consider any family (D_{α})_{α∈Λ} of subcoalgebras of C. Then I := ∑_{α} D_{α}^{⊥} is a two-sided ideal of C^{∗}, as a sum of two-sided ideals. Hence I^{⊥} is a subcoalgebra according to proposition I.3.2. But I^{⊥} is indeed the intersection of the subcoalgebras D_{α}.

In particular, the intersection of all subcoalgebras containing a given subset P of C will be called the subcoalgebra generated by P. We can now state the fundamental theorem of coalgebra theory :

Theorem I.3.6. Let C be a coalgebra. Then the subcoalgebra generated by one single element x is finitedimensional.

Proof. The coalgebra C is a left comodule over itself. Let N be the left subcomodule generated by x. According to theorem I.3.4, N is finite-dimensional. Then N^{⊥} has finite codimension, equal to dim N. It is a left ideal thanks to proposition I.3.1. The quotient space E = C^{∗}/N^{⊥} is a finite-dimensional left module over C^{⊥}. Let K be the annihilator of this left module. As kernel of the associated representation ρ : C^{∗} → End E it has clearly finite codimension, and it is a two-sided ideal.

Now K^{⊥} is a subcoalgebra according to proposition I.3.2. Moreover it is finitedimensional, as dim K^{⊥} = codim K^{⊥⊥} ≤ codim K. Finally K ⊂ N^{⊥} implies that N^{⊥⊥} ⊂ K^{⊥}. A fortiori N ⊂ K^{⊥}, so x belongs to K^{⊥}. The subcoalgebra generated by x is then included in the finite-dimensional subcoalgebra K^{⊥}, which proves the theorem.

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A coalgebra C is said to be irreducible if two nonzero subcoalgebras of C have always nonzero intersection. A simple coalgebra is a coalgebra which does not contain any proper subcoalgebra. A coalgebra C will be called pointed if any simple subcoalgebra of C is one-dimensional.

Lemma I.3.7. Any coalgebra C contains a simple subcoalgebra.

Proof. According to theorem I.3.6 we may suppose that C is finite-dimensional, and the lemma is immediate in this case.

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Proposition I.3.8. A coalgebra C is irreducible if and only if it contains a unique simple subcoalgebra.

Proof. Suppose C irreducible, and suppose that D_{1} and D_{2} are two simple subcoalgebras. The intersection D_{1} ∩ D_{2} is nonzero, and hence, by simplicity, D_{1} = D_{2}. Conversely suppose that E is the only simple subcoalgebra of C, and let D any subcoalgebra. According to lemma I.3.7 we have E ⊂ D, hence E is included in any intersection of subcoalgebras, which proves that C is irreducible.

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Lemma I.3.9. Let (C_{α})_{α∈Λ} a family of subcoalgebras of a coalgebra C such that C is the direct sum of the C_{α}’s. Then for any subcoalgebra D we have :

D = ⨁_{α∈Λ} D ∩ C_{α}.

Proof. The sum is indeed direct and included in D. To prove the reverse inclusion, pick any y in D and decompose it inside C :

y = ∑_{α∈Λ} y_{α}

with y_{α} ∈ C_{α} (finite sum). Consider for any γ ∈ Λ the linear form f_{γ} defined by :

\begin{aligned}f_{γ}|_{C_{α}} & = ε|_{C_{α}} si α = γ \\ & = 0 si α ≠ γ. \end{aligned}

Then f_{γ}(y) = ε(y_{γ}). Now we have :

(I ⊗ f_{γ}) ∘ Δ(y) = ∑_{α}(I ⊗ f_{γ}) ∘ Δ(y_{α})

= ∑_{α} ∑_{(y_{α})}(y_{α})_{1}f_{γ}((y_{α})_{2})

= ∑_{(y_{γ})}(y_{γ})_{1}ε((y_{γ})_{2})

= (I ⊗ ε) ∘ Δ(y_{γ})

= y_{γ}.

This shows that y_{γ} in in D, which proves the lemma.

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Let us define the coradical of a coalgebra C as the sum R of its simple subcoalgebras. As indicated by the terminology this notion is dual to the notion of Jacobson radical of an algebra : cf. proposition I.3.11 below.

Proposition I.3.10. Let R be the coradical of a coalgebra C. Then for any subcoalgebra D the coradical R_{D} of D is equal to R ∩ D.

Proof. Any simple subcoalgebra of D is a simple subcoalgebra of C, so R_{D} ⊂ R ∩ D. Conversely by lemma I.3.9 R ∩ D is a direct sum of simple subcoalgebras of D, hence R ∩ D ⊂ R_{D}.

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Proposition I.3.11. If C is a finite-dimensional coalgebra with coradical R, then R^{⊥} is the Jacobson radical of the algebra C^{∗}.

Proof. If S is a simple subcoalgebra of C it is clear from dimension considerations that S^{⊥} is a maximal two-sided ideal of C^{∗}. Conversely any maximal two-sided ideal of C^{∗} is the orthogonal of a simple subcoalgebra, so R^{⊥} is indeed the intersection of all maximal two-sided ideals of C^{∗}. Finally (lemma I.2.8), maximal two-sided and primitive ideals of a finite-dimensional algebra coincide.

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I.3.4. The wedge

Let C be a coalgebra and X, Y two linear subspaces of C. We define :

X ∧ Y = {x ∈ C, Δx ∈ X ⊗ C + C ⊗ Y }.

We define as well inductively :

∧^{0}X = {0}, ∧^{n}X = (∧^{n−1}X) ∧ X.

The alternative definition in terms of the algebra structure on C^{∗} is often more manageable, and its verification is straightforward :

X ∧ Y = (X^{⊥}Y^{⊥})^{⊥}.

Here is a first application of this definition :

Proposition I.3.12.

1) The wedge is associative : (X ∧ Y) ∧ Z = X ∧ (Y ∧ Z).

2) If X is a left coideal then {0} ∧ X = X.

3) If X is a left coideal and Y is a right coideal, then X ∧ Y is a subcoalgebra of C.

4) The wedge of two subcoalgebras is a subcoalgebra.

5) If X ⊂ X′ and Y ⊂ Y′, then X ∧ Y ⊂ X′ ∧ Y′.

Proof. According to the definition we have :

(X ∧ Y) ∧ Z = X ∧ (Y ∧ Z) = (X^{⊥}Y^{⊥}Z^{⊥})^{⊥}.

Let X (resp. Y) be a left (resp. right) coideal. According to proposition I.3.1, X^{⊥} is a left ideal of C^{∗} and Y^{⊥} is a right ideal. The product (X^{⊥}Y^{⊥}) is then a two-sided ideal. Second and third assertions follow by applying proposition I.3.1 again, and by noticing that :

{0} ∧ X = (C^{∗}.X^{⊥})^{⊥} = X^{⊥⊥} = X.

4) is an immediate consequence of 3), and 5) is clear.

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The wedge admits the following comodule version : let M be a right C-comodule with coaction Φ : M → M ⊗ C. Let N be a subspace of M and X be a subspace of C. We define :

N ∧ X = {x ∈ M, Φx ∈ N ⊗ C + C ⊗ X}.

One can check that if X is a right coideal the N ∧ X is a subcomodule, and if N is a subcomodule then N ⊂ N ∧ X.

Proposition I.3.13. Let R be the coradical of a coalgebra C, and let M be a right C-comodule. Then for {0} ⊂ M we have :

⋃_{n}{0} ∧ (∧^{n}M) = M.

Proof. Let x ∈ M, and let N be a finite-dimensional subcomodule containing x (which exists thanks to theorem I.3.4). If Φ denotes the coaction, then Φ(N) ⊂ N ⊗X where X is a finite-dimensional subspace of C. Let D be a finite-dimensional subcoalgebra containing X (which exists thanks to theorem I.3.6). It is clear that N is a right D-comodule.

Applying Proposition I.3.10, the coradical of D is R_{0} = R∩D. Proposition I.3.11 says that R_{0}^{⊥} is the Jacobson radical of D^{∗}, which is nilpotent by corollary I.2.7 : there exists a positive integer n such that (R_{0}^{⊥})^{n} = {0}. Dualizing we get that ∧_{D}^{n}R_{0} = D, where the subscribe D reminds with respect to which coalgebra the wedge operation is performed. Clearly we have :

∧_{D}^{n}R_{0} ⊂ ∧_{C}^{n}R_{0} ⊂ ∧_{C}^{n}R,

the second inclusion coming from assertion 5) of proposition I.3.12. We have then :

D ⊂ ∧^{n}R

(we have dropped the subscribe “C” here), hence the inclusions :

N ⊂ {0} ∧ D ⊂ {0} ∧ (∧^{n}R).

The initial element x belongs then to {0} ∧ (∧^{n}R) for some n, which proves the assertion.

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I.3.5. The coradical filtration

Let C be a coalgebra with coradical R. We consider for any integer i ≥ 0 :

C^{i} = ∧^{i+1}R.

The following proposition is an immediate consequence of propositions I.3.12 and I.3.13 :

Proposition I.3.14. (C^{i})_{i≥0} is an increasing sequence of subcoalgebras of C, and we have :

C = ⋃_{i≥0} C^{i}.

The coalgebra C is then endowed with an increasing filtration by subcoalgebras : its coradical filtration.

Proposition I.3.15. The coproduct is compatible with the coradical filtration, in the sense that the following inclusion holds :

Δ(C^{n}) ⊂ ∑_{i=0}^{n} C^{i} ⊗ C^{n−i}.

Proof. For any i ∈ {0, …, n + 1} we have :

∧^{n+1}R = (∧^{i}R) ∧ (∧^{n−i+1}R).

This is immediate for i = 1, …, n, and comes from proposition I.3.12 assertion 2) for i = 0 and i = n + 1. We have then (setting C^{−1} = {0}) :

ΔC^{n} ⊂ ⋂_{i=0}^{n+1}(C ⊗ C^{n−i} + C^{i−1} ⊗ C).

The right-hand side (RHS) is contained in C^{n} ⊗ C^{n}. Choose any supplementary subspace D_{i} of C^{i−1} inside C^{i} : so C^{0} = D_{0}, and C^{i} = D_{0} ⊕ ⋯ ⊕ D_{i}. We have :

(RHS) = ⋂_{i=0}^{n+1} ⨁_{r≤i−1 or s≤n−i} D_{r} ⊗ D_{s}

= ⨁_{r+s≤n} D_{r} ⊗ D_{s}

= ∑_{i=0}^{n} C^{i} ⊗ C^{n−i},

which proves the result.

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I.4. Convolution product

Let A be an algebra and C be a coalgebra (over the same field k). Then there is an associative product on L(C, A) called the convolution product. It is given by :

φ ∗ ψ = m_{A} ∘ (φ ⊗ ψ) ∘ Δ_{C}.

In Sweedler’s notation it reads :

φ ∗ ψ(x) = ∑_{(x)} φ(x_{1})ψ(x_{2}).

The associativity is a direct consequence of both associativity of A and coassociativity of C. We shall study this product more thoroughly in the next chapters.

I.5. Bialgebras and Hopf algebras

A (unital and co-unital) bialgebra is a vector space H endowed with a structure of unital algebra (m, u) and a structure of co-unital coalgebra (Δ, ε) which are compatible. The compatibility requirement is that Δ is an algebra morphism (or equivalently that m is a coalgebra morphism), ε is an algebra morphism and u is a coalgebra morphism. It is expressed by the commutativity of the three following diagrams :

\xymatrix{H ⊗ H ⊗ H ⊗ H \ar[rr]^{τ_{23}} &  & H ⊗ H ⊗ H ⊗ H \ar[d]^{m⊗m} \\ H ⊗ H \ar[u]_{Δ⊗Δ} \ar[r]_{m} & H \ar[r]_{Δ} & H ⊗ H}

\xymatrix{H ⊗ H \ar[d]^{m} \ar[r]^{ε⊗ε} & k ⊗ k \ar[d]^{∼} & H ⊗ H & k ⊗ k \ar[l]_{u⊗u} \\ H \ar[r]^{ε} & k & H \ar[u]_{Δ} & k \ar[l]_{u} \ar[u]_{∼}}

A Hopf algebra is a bialgebra H together with a linear map S : H → H called the antipode, such that the following diagram commutes :

\xymatrix{ & H ⊗ H \ar[rr]^{S⊗I} &  & H ⊗ H \ar[dr]^{m} \\ H \ar[rr]^{ε} \ar[dr]^{Δ} \ar[ur]^{Δ} &  & k \ar[rr]^{u} &  & H \\  & H ⊗ H \ar[rr]^{I⊗S} &  & H ⊗ H \ar[ur]^{m}}

In Sweedler’s notation it reads :

∑_{(x)} S(x_{1})x_{2} = ∑_{(x)} x_{1}S(x_{2}) = (u ∘ ε)(x).

In other words the antipode is an inverse of the identity I for the convolution product on L(H, H). The unit for the convolution is the map u ∘ ε.

Let H be a bialgebra. A primitive element in H is an element x such that Δx = x ⊗ 1 + 1 ⊗ x. A grouplike element is a nonzero element x such that Δx = x ⊗ x. Note that grouplike elements make sense in any coalgebra.

I.6. Examples

I.6.1. The Hopf algebra of a group

Let G be a group, and let kG be the group algebra (over the field k). It is by definition the vector space freely generated by the elements of G : the product of G extends uniquely to a bilinear map from kG×kG into kG, hence a multiplication m : kG⊗kG → kG, which is associative. The neutral element of G gives the unit for m.

The space kG is also endowed with a co-unital coalgebra structure, given by :

Δ(∑ λ_{i}g_{i}) = ∑ λ_{i}.g_{i} ⊗ g_{i}

and :

ε(∑ λ_{i}g_{i}) = ∑ λ_{i}.

This defines the coalgebra of the set G : it does not take into account the extra group structure on G, as the algebra structure does.

Proposition I.6.1. The vector space kG endowed with the algebra and coalgebra structures defined above is a Hopf algebra. The antipode is given by :

S(g) = g^{−1}, g ∈ G.

Proof. The compatibility of the product and the coproduct is an immediate consequence of the following computation : for any g, h ∈ G we have :

Δ(gh) = gh ⊗ gh = (g ⊗ g)(h ⊗ h) = Δg.Δh.

Now m(S ⊗ I)Δ(g) = g^{−1}g = e and similarly for m(I ⊗ S)Δ(g). But e = u ∘ ε(g) for any g ∈ G, so map S is indeed the antipode.

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Remark : if G were only a semigroup, the same construction would lead to a bialgebra structure on kG : the Hopf algebra structure (i.e. the existence of an antipode) reflects the group structure (the existence of the inverse). We have S^{2} = I in this case, but involutivity of the antipode is not true for general Hopf algebras.

I.6.2. Tensor algebras

There is a natural structure of cocommutative Hopf algebra on the tensor algebra T (V) of any vector space V. Namely we define the coproduct Δ as the unique algebra morphism from T (V) into T (V) ⊗ T (V) such that :

Δ(1) = 1 ⊗ 1, Δ(x) = x ⊗ 1 + 1 ⊗ x, x ∈ V.

We define the co-unit as the algebra morphism such that ε(1) = 1 and ε|_{V} = 0 This endows T (V) with a cocommutative bialgebra structure. We claim that the principal anti-automorphism :

S(x_{1} ⊗ ⋯ ⊗ x_{n}) = (−1)^{n}x_{n} ⊗ ⋯ ⊗ x_{1}

verifies the axioms of an antipode, so that T (V) is indeed a Hopf algebra. For x ∈ V we have S(x) = −x, hence S ∗ I(x) = I ∗ S(x) = 0. As V generates T (V) as an algebra it is easy to conclude.

I.6.3. Enveloping algebras

Let 𝔤 a Lie algebra. The universal enveloping algebra is the quotient of the tensor algebra T (𝔤) by the ideal J generated by x ⊗ y − y ⊗ x − [x, y], x, y ∈ 𝔤.

Lemma I.6.2. J is a Hopf ideal, i.e. Δ(J) ⊂ H ⊗ J + J ⊗ H and S(H) ⊂ H.

Proof. The last assertion is immediate. The first comes easily from the fact that the ideal J is generated by primitive elements (according to proposition I.7.3 below) : indeed any ideal generated by primitive elements is a Hopf ideal (very easy and left to the reader).

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The quotient of a Hopf algebra by a Hopf ideal is a Hopf algebra. Hence the universal enveloping algebra U(𝔤) is a cocommutative Hopf algebra.

I.7. Some properties of Hopf algebras

We summarize in the proposition below the main properties of the antipode in a Hopf algebra :

Proposition I.7.1 (cf. [Sw] proposition 4.0.1). Let H be a Hopf algebra with multiplication m, comultiplication Δ, unit u : 1 ↦ 1, co-unit ε and antipode S. Then :

1) S ∘ u = u and ε ∘ S = ε.

2) S is an algebra antimorphism and a coalgebra antimorphism, i.e. if τ denotes the flip we have :

m ∘ (S ⊗ S) ∘ τ = S ∘ m, τ ∘ (S ⊗ S) ∘ Δ = Δ ∘ S.

3) If H is commutative or cocommutative, then S^{2} = I.

Proof. We follow closely the proof given by Chr. Kassel here [K]. Starting from ε(1) = 1 and Δ(1) = 1 ⊗ 1, we get : 1 = m(S ⊗ I)Δ(1) = S(1)1, so S(1) = 1. We have for any x ∈ H :

ε ∘ u ∘ ε(x) = ε(ε(x).1) = ε(x).

Then,

ε(x) = ε(m(S ⊗ I)Δ)(x)

= ε(∑_{(x)} S(x_{1})x_{2})

= ∑_{(x)}(ε ∘ S)(x_{1})ε(x_{2}).

On the other hand,

S(x) = S ∗ (u ∘ ε)(x) = ∑_{(x)} S(x_{1})(u ∘ ε)(x_{2}).

Hence,

(ε ∘ S)(x) = ∑_{(x)}(ε ∘ S)(x_{1})ε(x_{2}).

This proves assertion 1). Now consider m, N, P ∈ L(H ⊗ H, H) defined by :

m(x ⊗ y) = xy,  N(x ⊗ y) = S(y)S(x),  P (x ⊗ y) = S(xy).

Considering the convolution product ˜∗ on L(H ⊗ H, H) we shall prove :

Lemma I.7.2.

P˜∗m = u ∘ ε_{H⊗H} = m˜∗N.

Proof. We compute, with Sweedler’s notation :

P˜∗m(x ⊗ y) = ∑_{(x⊗y)} P ((x ⊗ y)_{1})m((x ⊗ y)_{2})

= ∑_{(x),(y)} P (x_{1} ⊗ y_{1})m(x_{2} ⊗ y_{2})

= ∑_{(x),(y)} S(x_{1}y_{1})x_{2}y_{2}

= (S ∗ I)(xy) = (u ∘ ε)(xy) = (u ∘ ε_{H⊗H})(x ⊗ y), and :

m˜∗N(x ⊗ y) = ∑_{(x),(y)} m(x_{1} ⊗ y_{1})N(x_{2} ⊗ y_{2})

= ∑_{(x),(y)} x_{1}y_{1}S(y_{2})S(x_{2})

= ∑_{(x)} x_{1}(u ∘ ε)(y)S(x_{2})

= (u ∘ ε)(x)(u ∘ ε)(y) = (u ∘ ε_{H⊗H})(x ⊗ y),

which proves the lemma.

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End of proof of Proposition I.7.1 : as u ∘ ε_{H⊗H} is the unit element for ˜∗ lemma proves that both P and N are inverse of m for this convolution. Hence :

P = P˜∗(m˜∗N) = (P˜∗m)˜∗N = N,

which proves the first part of assertion 2). The second part is proved similarly using convolution in L(H, H ⊗ H) with m replaced with Δ, N replaced with τ ∘ (S ⊗ S) ∘ Δ and P replaced with Δ ∘ S. Indeed,

((Δ ∘ S)˜∗Δ)(x) = (m ⊗ m) ∘ τ_{23} ∘ ((Δ ∘ S) ⊗ Δ) ∘ Δ(x)
= (m ⊗ m) ∘ τ_{23} ∘ (Δ ⊗ Δ) ∘ (S ⊗ I) ∘ Δ(x)
= Δ ∘ m ∘ (S ⊗ I) ∘ Δ(x)
= Δ(u ∘ ε(x))
= u ∘ ε(x) ⊗ u ∘ ε(x),

and :

Δ˜∗(τ ∘ (S ⊗ S) ∘ Δ)(x) = (m ⊗ m) ∘ τ_{23} ∘ (Δ ⊗ (τ ∘ (S ⊗ S) ∘ Δ)) ∘ Δ(x)

= (m ⊗ m) ∘ τ_{23} ∘ τ_{34} ∘ (I ⊗ I ⊗ S ⊗ S) ∘ (Δ ⊗ Δ) ∘ Δ(x)

= (m ⊗ m) ∘ τ_{23} ∘ τ_{34} ∘ (I ⊗ I ⊗ S ⊗ S) ∘ (I ⊗ Δ ⊗ I) ∘ (Δ ⊗ I) ∘ Δ(x)

= (m ⊗ m) ∘ τ_{23} ∘ τ_{34} ∘ (I ⊗ I ⊗ S ⊗ S) ∘ (I ⊗ Δ ⊗ I)(∑_{(x)} x_{1} ⊗ x_{2} ⊗ x_{3})

= (m ⊗ m) ∘ τ_{23} ∘ τ_{34} ∘ (∑_{(x)} x_{1} ⊗ x_{2:1} ⊗ Sx_{2:2} ⊗ Sx_{3})

= ∑_{(x)} x_{1}Sx_{3} ⊗ x_{2:1}Sx_{2:2}

= ∑_{(x)} x_{1}Sx_{3} ⊗ (u ∘ ε)(x_{2})

= (∑_{(x)} x_{1}Sx_{2}) ⊗ (u ∘ ε)(x)

= u ∘ ε(x) ⊗ u ∘ ε(x).

If H is commutative then :

u ∘ ε(x) = ∑_{(x)} x_{2}S(x_{1}).

If H is cocommutative, this is again true, as :

u ∘ ε(x) = (I ∗ S)(x) = ∑_{(x)} x_{1}S(x_{2}).

Suppose that H is commutative or cocommutative. Then using assertion 2) we can compute :

(S ∘ S) ∗ S(x) = ∑_{(x)}(S ∘ S)(x_{1})S(x_{2})

= S(∑_{(x)} x_{2}S(x_{1}))

= S(u ∘ ε(x))

= (u ∘ ε)(x).

Thus S ∘ S is a left inverse for S (for the convolution), and then S ∘ S = I. This ends the proof of Proposition I.7.1.

•

Proposition I.7.3. 1). If x is a primitive element then S(x) = −x. 2). The linear subspace Prim H of primitive elements in H is a Lie algebra.

Proof. If x is primitive, then (ε⊗ε)∘Δ(x) = 2ε(x). On the other hand, (ε⊗ε)∘Δ(x) = ε(x), so ε(x) = 0. Then :

0 = (u ∘ ε)(x) = m(S ⊗ I)Δ(x) = S(x) − x.

Now let x and y be primitive elements of H. Then we can easily compute :

Δ(xy − yx) = (x ⊗ 1 + 1 ⊗ x)(y ⊗ 1 + 1 ⊗ y) − (y ⊗ 1 + 1 ⊗ y)(x ⊗ 1 + 1 ⊗ x)
= (xy − yx) ⊗ 1 + 1 ⊗ (xy + yx) + x ⊗ y + y ⊗ x − y ⊗ x − x ⊗ y
= (xy − yx) ⊗ 1 + 1 ⊗ (xy − yx).
•

II. Convolution product and regularization

II.1. Connected graded bialgebras

Let k be a field with vanishing characteristic. We shall denote by k[[t]] the ring of formal series on k, and by k[t^{−1}, t]] the field of Laurent series on k. A graded Hopf algebra on k is a graded k-vector space :

H = ⨁_{n≥0} H_{n}

endowed with a product m : H ⊗ H → H, a coproduct Δ : H → H ⊗ H, a unity u : k → H, a co-unity ε : H → k and an antipode S : H → H fulfilling the usual axioms of a Hopf algebra, and such that :

\begin{aligned}H_{p}.H_{q} & ⊂ H_{p+q} \\ Δ(H_{n}) & ⊂ ⨁_{p+q=n} H_{p} ⊗ H_{q}. \\ S(H_{n}) & ⊂ H_{n} \end{aligned}

If we do not ask for the existence of an antipode H we get the definition of a graded bialgebra. In a graded bialgebra H we shall consider the increasing filtration :

H^{n} = ⨁_{p=0}^{n} H_{p}.

Suppose moreover that H is connected, i.e. H_{0} is one-dimensional. Then we have :

Ker ε = ⨁_{n≥1} H_{n}.

Proposition II.1.1. For any x ∈ H^{n}, n ≥ 1 we can write :

Δx = x ⊗ 1 + 1 ⊗ x + Δ̃x,  Δ̃x ∈ ⨁_{p+q=n,p≠0,q≠0} H_{p} ⊗ H_{q}.

The map Δ̃ is coassociative on Ker ε and Δ̃_{k} = (I^{⊗k−1} ⊗Δ̃)(I^{⊗k−2} ⊗Δ̃)...Δ̃ sends H^{n} into (H^{n−k})^{⊗k+1}.

Proof. Thanks to connectedness we clearly can write :

Δx = a(x ⊗ 1) + b(1 ⊗ x) + Δ̃x

with a, b ∈ k and Δ̃x ∈ Ker ε ⊗ Ker ε. The co-unity property then tells us that, with k ⊗ H and H ⊗ k canonically identified with H :

\begin{aligned}x & = (ε ⊗ I)(Δx) = bx \\ x & = (I ⊗ ε)(Δx) = ax, \end{aligned}

hence a = b = 1. We shall use the following two variants of Sweedler’s notation :

\begin{aligned}Δx & = ∑_{(x)} x_{1} ⊗ x_{2}, \\ Δ̃x & = ∑_{(x)} x′ ⊗ x′′, \end{aligned}

the second being relevant only for x ∈ Ker ε. if x is homogeneous of degree n we can suppose that the components x_{1}, x_{2}, x′, x′′ in the expressions above are homogeneous as well, and we have then |x_{1}| + |x_{2}| = n and |x′| + |x′′| = n We easily compute :

(Δ ⊗ I)Δ(x) = x ⊗ 1 ⊗ 1 + 1 ⊗ x ⊗ 1 + 1 ⊗ 1 ⊗ x

+ ∑_{(x)} x′ ⊗ x′′ ⊗ 1 + x′ ⊗ 1 ⊗ x′′ + 1 ⊗ x′ ⊗ x′′

+ (Δ̃ ⊗ I)Δ̃(x)

and

(I ⊗ Δ)Δ(x) = x ⊗ 1 ⊗ 1 + 1 ⊗ x ⊗ 1 + 1 ⊗ 1 ⊗ x

+ ∑_{(x)} x′ ⊗ x′′ ⊗ 1 + x′ ⊗ 1 ⊗ x′′ + 1 ⊗ x′ ⊗ x′′

+ (I ⊗ Δ̃)Δ̃(x),

hence the co-associativity of Δ̃ comes from the one of Δ. Finally it is easily seen by induction on k that for any x ∈ H^{n} we can write :

Δ̃_{k}(x) = ∑_{x} x^{(1)} ⊗ ⋯ ⊗ x^{(k+1)},

with |x^{(j)}| ≥ 1. The grading imposes :

∑_{j=1}^{k+1} |x^{(j)}| = n,

so the maximum possible for any degree |x^{(j)}| is n − k.

•

II.2. Connected filtered bialgebras

A filtered Hopf algebra on k is a k-vector space together with an increasing ℤ_{+}-indexed filtration :

H^{0} ⊂ H^{1} ⊂ ⋯ ⊂ H^{n} ⊂ ⋯, ⋃_{n} H^{n} = H

endowed with a product m : H⊗H → H, a coproduct Δ : H → H⊗H, a unit characteristic u : k → H, a co-unit ε : H → k and an antipode S : H → H fulfilling the usual axioms of a Hopf algebra, and such that :

\begin{aligned}H^{p}.H^{q} & ⊂ H^{p+q} \\ Δ(H^{n}) & ⊂ ∑_{p+q=n} H^{p} ⊗ H^{q} \\ S(H^{n}) & ⊂ H^{n}. \end{aligned}

If we do not ask for the existence of an antipode H we get the definition of a filtered bialgebra. For any x ∈ H we set :

|x| = min{n ∈ ℕ, x ∈ H^{n}}.

Any graded bialgebra or Hopf algebra is obviously filtered by the canonical filtration associated to the grading :

H^{n} = ⨁_{i=0}^{n} H_{i},

and in that case, if x is an homogeneous element, x is of degree n if and only if |x| = n. We say that the filtered bialgebra H is connected if H^{0} is one-dimensional. There is an analogue of proposition II.1.1 in the connected filtered case :

Proposition II.2.1. For any x ∈ H^{n}, n ≥ 1 we can write :

Δx = x ⊗ 1 + 1 ⊗ x + Δ̃x,  Δ̃x ∈ ∑_{p+q=n,p≠0,q≠0} H^{p} ⊗ H^{q}.

The map Δ̃ is coassociative on Ker ε and Δ̃_{k} = (I^{⊗k−1} ⊗Δ̃)(I^{⊗k−2} ⊗Δ̃)...Δ̃ sends H^{n} into (H^{n−k})^{⊗k+1}.

Proof. Straightforward adaptation of proof of proposition II.1.1.

•

The following theorem is due to S. Montgomery [Mo lemma 1.1].

Theorem II.2.2. Let H be any pointed Hopf algebra. Then the coradical filtration endows H with a structure of filtered Hopf algebra.

Proof. It only remains to show that for any n ∈ ℕ the inclusion S(H^{n}) ⊂ H^{n} holds, and that for any p, q ∈ ℕ :

H^{p}H^{q} ⊂ H^{p+q},

which, together with proposition I.3.15, will prove the result. Recall that a pointed coalgebra is a coalgebra in which any simple subcoalgebra is one-dimensional. In this case any simple subcoalgebra is linearly generated by a unique grouplike element. Any grouplike element g in a Hopf algebra admits an inverse Sg, where S is the antipode. It follows that the coradical H^{0} of a pointed Hopf algebra H is a Hopf subalgebra of H, precisely the Hopf algebra of the group of the grouplike elements of H (cf. example I.6.1).

The proof proceeds by induction : inclusion SH_{0} ⊂ H_{0} obviously holds. Suppose that SH^{k} ⊂ H^{k} for all k ≤ n − 1. Using the definition of H^{n} :

H^{n} = H^{0} ∧ H^{n−1} = H^{n−1} ∧ H^{0}

and the formula :

Sx = ∑_{(x)} Sx_{2} ⊗ Sx_{1},

(cf. proposition I.7.1) we deduce the inclusion SH^{n} ⊂ H^{n}. Now, suppose that the inclusion H^{k}H^{0} ⊂ H^{k} holds for k ≤ n − 1 (its is obviously the case for k = 0). Then we have :

Δ(H^{n}H^{0}) ⊂ (H^{0} ⊗ H + H ⊗ H^{n−1})(H^{0} ⊗ H^{0})
⊂ H^{0} ⊗ H + H ⊗ H^{n−1}H^{0}
⊂ H^{0} ⊗ H + H ⊗ H^{n−1}.

So H^{n}H^{0} ⊂ Δ^{−1}(H^{0} ⊗ H + H ⊗ H^{n−1}) = H^{n}. Similarly on the other side we have H^{0}H^{n} ⊂ H^{n} for any n. Suppose now that the inclusion :

H^{p}H^{q} ⊂ H^{p+q}

holds for any p, q such that p + q ≤ n − 1. Choose p, q with p + q = n and compute :

Δ(H^{p}H^{q}) ⊂ (H^{0} ⊗ H^{p} + H^{p} ⊗ H^{p−1})(H^{0} ⊗ H^{q} + H^{q} ⊗ H^{q−1})

⊂ H^{0}H^{0} ⊗ H^{p}H^{q} + H^{p}H^{0} ⊗ H^{p−1}H^{q} + H^{0}H^{q} ⊗ H^{p}H^{q−1} + H^{p}H^{q} ⊗ H^{p−1}H^{q−1}

⊂ H^{0} ⊗ H + H ⊗ H^{p+q−1}

thanks to the induction hypothesis and the property already proved when one of the indices is equal to zero. Thus H^{p}H^{q} ⊂ H^{p+q}, which finishes the proof of the theorem.

•

Remark 1 : The proof used only the property that the coradical is a Hopf subalgebra of H. The pointedness of H implies this property but is not strictly necessary.

Remark 2 : the image of k under the unit map u is a one-dimensional simple subcoagebra of H. If H is an irreducible coalgebra, by proposition I.3.8 it is the unique one, and then the coradical is H^{0} = k.1. Any irreducible Hopf algebra is then pointed, and connected with respect to the coradical filtration.

II.3. The convolution product

An important result is that any connected filtered bialgebra is indeed a filtered Hopf algebra, in the sense that the antipode comes for free. We give a proof of this fact as well as a recursive formula for the antipode with the help of the convolution product :

Let H be a (connected filtered) bialgebra, and let A be any k-algebra : the convolution product on L(H, A) is given by :

\begin{aligned}φ ∗ ψ(x) & = m_{A}(φ ⊗ ψ)Δ(x) \\ & = ∑_{(x)} φ(x_{1})ψ(x_{2}). \end{aligned}

Proposition II.3.1. The map e = u_{A} ∘ ε, given by e(1) = 1_{A} and e(x) = 0 for any x ∈ Ker ε, is a unit for the convolution product. Moreover the set G = {φ ∈ L(H, A), φ(1) = 1_{A}} endowed with the convolution product is a group.

Proof. The first statement is straightforward. To prove the second let us consider the formal series :

\begin{aligned}φ^{∗−1}(x) & = (e − (e − φ))^{∗−1}(x) \\ & = ∑_{k≥0}(e − φ)^{∗k}(x). \end{aligned}

Using (e − φ)(1) = 0 we have immediately (e − φ)^{∗k}(1) = 0, and for any x ∈ Ker ε :

(e − φ)^{∗k}(x) = m_{A,k−1}(φ ⊗ ⋯ ⊗ φ)Δ̃_{k−1}(x).

When x ∈ H^{n} this expression vanishes then for k ≥ n + 1. The formal series ends up then with a finite number of terms for any x, which proves the result.

•

Corollary II.3.2. Any connected filtered bialgebra H is a filtered Hopf algebra. The antipode is defined by :

S(x) = ∑_{k≥0}(uε − I)^{∗k}(x).

It is given by S(1) = 1 and recursively by any of the two formulas for x ∈ Ker ε :

\begin{aligned}S(x) & = −x − ∑_{(x)} S(x′)x′′ \\ S(x) & = −x − ∑_{(x)} x′S(x′′). \end{aligned}

Proof. The antipode, when it exists, is the inverse of the identity for the convolution product on L(H, H). One just needs then to apply proposition II.3.1 with A = H. The two recursive formulas come directly from the two equalities :

m(S ⊗ I)Δ(x) = m(I ⊗ S)Δ(x) = 0

fulfilled by any x ∈ Ker ε.

•

Let 𝔤 be the subspace of L(H, A) formed by the elements α such that α(1) = 0. It is clearly a subalgebra of L(H, A) for the convolution product. We have :

G = e + 𝔤.

From now on we shall suppose that the ground field k is of characteristic zero. For any x ∈ H^{n} the exponential :

e^{∗α}(x) = ∑_{k≥0} \frac{α^{∗k}(x)}{k!}

is a finite sum (ending up at k = n). It is a bijection from 𝔤 onto G. Its inverse is given by :

Log(1 + α)(x) = ∑_{k≥1} \frac{(−1)^{k−1}}{k} α^{∗k}(x).

This sum again ends up at k = n for any x ∈ H^{n}. Let us introduce a decreasing filtration on L = L(H, A) :

L^{n} = {α ∈ L, α|_{H^{n−1}} = 0}.

Clearly L_{0} = L and L_{1} = 𝔤. We define the valuation val φ of an element φ of L as the biggest integer k such that φ is in L_{k}. We shall consider in the sequel the ultrametric distance on L induced by the filtration :

d(φ, ψ) = 2^{− val(φ−ψ)}.

For any α, β ∈ 𝔤 let [α, β] = α ∗ β − β ∗ α.

Proposition II.3.3.

1) We have the inclusion :

L^{p} ∗ L^{q} ⊂ L^{p+q}.

2) The metric space L endowed with then distance defined just above is complete.

Proof. Take any x ∈ H^{p+q−1}, and any α ∈ L_{p} and β ∈ L_{q}. We have :

(α ∗ β)(x) = ∑_{(x)} α(x_{1})β(x_{2}).

Recall that we denote by |x| the minimal n such that x ∈ H^{n}. Since |x_{1}| + |x_{2}| = |x| ≤ p + q − 1, either |x_{1}| ≤ p − 1 or |x_{2}| ≤ q − 1, so the expression vanishes. Now if (ψ_{n}) is a Cauchy sequence in L it is immediate to see that this sequence is locally stationary, i.e. for any x ∈ H there exists N(x) ∈ ℕ such that ψ_{n}(x) = ψ_{N(x)}(x) for any n ≥ N(x). Then the limit of (ψ_{n}) exists and is clearly defined by :

ψ(x) = ψ_{N(x)}(x).
•

As a corollary the Lie algebra L_{1} = 𝔤 is pro-nilpotent, in a sense that it is the projective limit of the Lie algebras 𝔤/L^{n}, which are nilpotent.

II.4. Algebra morphisms and cocycles

Let H be a connected filtered Hopf algebra over k, and let A be a k-algebra. A cocycle from H to A is a linear morphism τ : H → A such that τ(xy) = τ(yx) for any x, y ∈ H. It is indeed a 1-cocycle in the cohomology of the Lie algebra H with values in A considered as a trivial H-module. In the case where A is the ground field k cocycles are just traces.

We shall also consider algebra morphisms from H to A. When the algebra A is commutative we shall call them slightly abusively characters. It is clear that any character in our sense is a cocycle. We recover of course the usual notion of character when the algebra A is the ground field k.

The notions of character and cocycle involve only the algebra structure of H. Let us consider now the full Hopf algebra structure and see what happens with the convolution product :

Proposition II.4.1. Let H be a connected filtered Hopf algebra over k, and let A be a k-algebra. Then,

1). The convolution of two cocycles in L(H, A) is a cocycle.

2). If τ is a cocycle such that τ(1) = 1_{A}, then the inverse τ^{∗−1} is a cocycle as well.

3). In the case of a commutative algebra A the characters from H to A form a group Γ under the convolution product, and for any χ ∈ Γ the inverse is given by :

χ^{∗−1} = χ ∘ S.

Proof. Using the fact that Δ is an algebra morphism we have for any x, y ∈ H :

f ∗ g(xy) = ∑_{(x)(y)} f(x_{1}y_{1})g(x_{2}y_{2}).

If f and g are cocycles we get :

\begin{aligned}f ∗ g(xy) & = ∑_{(x)(y)} f(y_{1}x_{1})g(y_{2}x_{2}) \\ & = f ∗ g(yx). \end{aligned}

If A is commutative and if f and g are characters we get :

f ∗ g(xy) = ∑_{(x)(y)} f(x_{1})f(y_{1})g(x_{2})g(y_{2})

= ∑_{(x)(y)} f(x_{1})g(x_{2})f(y_{1})g(y_{2})

= (f ∗ g)(x)(f ∗ g)(y).

The unit e = u_{A}ε is both a cocycle and an algebra morphism. The formula for the inverse of a character comes easily from the commutativity of the following diagram :

\xymatrix{ & H ⊗ H \ar[rr]^{S⊗I} &  & H ⊗ H \ar[dr]^{m} \ar[rr]^{f⊗f} &  & ⊗A \ar[dr]^{m_{A}} \\ H \ar[rr]_{ε} \ar[dr]^{Δ} \ar[ur]^{Δ} \ar@{-->}[rrrrrr]_{f∗(f∘S)} \ar@{-->}[rrrrrr]^{(f∘S)∗f} \ar@{-->}[rrrrrr]^{e} &  & k \ar[rr]_{u} &  & H \ar[rr]_{f} &  & A \\  & H ⊗ H \ar[rr]^{I⊗S} &  & H ⊗ H \ar[ur]^{m} \ar[rr]^{f⊗f} &  & A ⊗ A \ar[ur]^{m_{A}}}

Finally the fact that the inverse of a cocycle τ such that τ(1) = 1_{A} is a cocycle comes from 1) and from the formula :

τ^{−1}(x) = ∑_{k≥0}(e − τ)^{∗k}(x).

•

We call derivations (or infinitesimal characters) with values in the algebra A those elements α of L(H, A) such that :

α(xy) = e(x)α(y) + α(x)e(y).

Proposition II.4.2. Suppose that A is a commutative algebra. Let G_{1} (resp. 𝔤_{1}) be the set of characters of H with values in A (resp the set of derivations of H with values in A), and let G_{2} (resp. 𝔤_{2}) be the set of cocycles φ from H to A such that φ(1) = 1_{A} (resp. φ(1) = 0). Then G_{1} and G_{2} are subgroups of G, the exponential restricts to a bijection from 𝔤_{1} onto G_{1} (resp. from 𝔤_{2} onto G_{2}), and 𝔤_{1}, 𝔤_{2} are Lie subalgebras of 𝔤.

Proof. Part of these results are a reformulation of proposition II.4.1 and some points are straightforward. The only non-trivial point concerns 𝔤_{1} and G_{1}. Take two derivations α and β with values in A and compute :

(α ∗ β)(xy) = ∑_{(x)(y)} α(x_{1}x_{2})β(y_{1}y_{2})

= ∑_{(x)(y)}(α(x_{1})e(y_{1}) + e(x_{1})α(y_{1})).(β(x_{2})e(y_{2}) + e(x_{2})α(y_{2}))

= (α ∗ β)(x)e(y) + α(x)β(y) + β(x)α(y) + e(x)(α ∗ β)(y).

Using the commutativity of A we immediately get :

[α, β](xy) = [α, β](x)e(y) + e(x)[α, β](y),

which shows that 𝔤_{1} is a Lie algebra. Now for α ∈ 𝔤_{1} we have :

α^{∗n}(xy) = ∑_{k=0}^{n} \binom{n}{k}α^{∗k}(x)α^{∗(n−k)}(y),

as easily see by induction on n. A straightforward computation then yields :

e^{∗α}(xy) = e^{∗α}(x)e^{∗α}(y).
•

II.5. Birkhoff decomposition

We consider here the situation where the algebra A admits a renormalization scheme, i.e. a splitting into two subalgebras :

A = A_{−} ⊕ A_{+}

with 1 ∈ A_{+}. As an example, take A as the field k[t^{−1}, t]] of Laurent series, A_{−} = t^{−1}k[t^{−1}] and A_{+} = k[[t]]. The projection on A_{−} parallel to A_{+} will be denoted by π.

Theorem II.5.1. 1). Let H be a connected filtered Hopf algebra. Let G be the group of those φ ∈ L(H, A) such that φ(1) = 1_{A} endowed with the convolution product. Any φ ∈ G admits a unique Birkhoff decomposition :

φ = φ_{−}^{∗−1} ∗ φ_{+},

where φ_{−} sends 1 to 1_{A} and Ker ε into A_{−}, and where φ_{+} sends H into A_{+}. The maps φ_{−} and φ_{+} are given on Ker ε by the following recursive formulas :

\begin{aligned}φ_{−}(x) & = −π(φ(x) + ∑_{(x)} φ_{−}(x′)φ(x′′)) \\ φ_{+}(x) & = (I − π)(φ(x) + ∑_{(x)} φ_{−}(x′)φ(x′′)). \end{aligned}

2). If τ ∈ G is a cocycle, the components τ_{−} and τ_{+} occurring in the Birkhoff decomposition of τ are cocycles as well.

3). If the algebra A is commutative and if χ is a character, the components χ_{−} and χ_{+} occurring in the Birkhoff decomposition of χ are characters as well.

Proof. Points 1) and 3) together give an abstract counterpart of Theorem 4 of [CK], point 2) is new up to my knowledge. The proof goes along the same lines : for the first assertion it is immediate from the definition of π that φ_{−} sends Ker ε into A_{−}, and that φ_{+} sends Ker ε into A_{+}. It only remains to check equality φ_{+} = φ_{−} ∗ φ, which is an easy computation :

φ_{+}(x) = (I − π)(φ(x) + ∑_{(x)} φ_{−}(x′)φ(x′′)).

= φ(x) + φ_{−}(x) + ∑_{(x)} φ_{−}(x′)φ(x′′)

= (φ_{−} ∗ φ)(x).

To prove the second assertion it is sufficient to prove that τ_{−} is a cocycle whenever τ is a cocycle. The same property for τ_{+} comes then from proposition II.3.1. We prove the formula τ_{−}(xy) = τ_{−}(yx) by induction on the integer d = |x| + |y| : it is true for d ≤ 1. Suppose the formula is true up to d−1 and take any x, y ∈ H with |x|+|y| = d. Decompose Δ(xy) with the second version of Sweedler’s notation :

Δ(xy) = xy ⊗ 1 + 1 ⊗ xy + x ⊗ y + y ⊗ x

+ ∑_{(x)}(x′y ⊗ x′′ + x′ ⊗ x′′y) + ∑_{(y)}(xy′ ⊗ y′′ + y′ ⊗ xy′′)

+ ∑_{(x)(y)} x′y′ ⊗ x′′y′′.

We have then :

τ_{−}(xy) = −π(τ(xy) + τ_{−}(x)τ(y) + τ_{−}(y)τ(x)

+ ∑_{(x)}(τ_{−}(x′y)τ(x′′) + τ_{−}(x′)τ(x′′y)) + ∑_{(y)}(τ_{−}(xy′)τ(y′′) + τ_{−}(y′)τ(xy′′))

+ ∑_{(x)(y)} τ_{−}(x′y′)τ(x′′y′′)),

whereas :

τ_{−}(yx) = −π(τ(yx) + τ_{−}(y)τ(x) + τ_{−}(x)τ(y)

+ ∑_{(y)}(τ_{−}(y′x)τ(y′′) + τ_{−}(y′)τ(y′′x)) + ∑_{(x)}(τ_{−}(yx′)τ(x′′) + τ_{−}(x′)τ(yx′′))

+ ∑_{(x)(y)} τ_{−}(y′x′)τ(y′′x′′)).

Using the cocycle property for τ and the induction hypothesis we see that the two expressions are the same.

The proof of assertion 3) goes exactly as in [CK] and relies on the following Rota-Baxter equality in A :

π(a)π(b) = −π(ab) + π(π(a)b) + π(π(b)a),

which is easily verified by decomposing a and b into their A_{±}-parts. Let χ be a character of H with values in A. Suppose that we have χ_{−}(xy) = χ_{−}(x)χ_{−}(y) for any x, y ∈ H such that |x| + |y| ≤ d − 1, and compute for x, y such that |x| + |y| = d :

χ_{−}(x)χ_{−}(y) = π(X)π(Y),

with X = χ(x)−∑_{(x)} χ_{−}(x′)χ(x′′) and Y = χ(y)−∑_{(y)} χ_{−}(y′)χ(y′′). Using the formula :

π(X) = −χ_{−}(x),

we get :

χ_{−}(x)χ_{−}(y) = −π(XY + χ_{−}(x)Y + Xχ_{−}(y)),

hence :

χ_{−}(x)χ_{−}(y) = −π(χ(x)χ(y) + χ_{−}(x)χ(y) + χ(x)χ_{−}(y)

+ ∑_{(x)} χ_{−}(x′)χ(x′′)(χ(y) + χ_{−}(y)) + ∑_{(y)}(χ(x) + χ_{−}(x))χ_{−}(y′)χ(y′′)

+ ∑_{(x)(y)} χ_{−}(x′)χ(x′′)χ_{−}(y′)χ(y′′)).

We have to compare this expression with :

χ_{−}(xy) = −π(χ(xy) + χ_{−}(x)χ(y) + χ_{−}(y)χ(x)

+ ∑_{(x)}(χ_{−}(x′y)χ(x′′) + χ_{−}(x′)χ(x′′y)) + ∑_{(y)}(χ_{−}(xy′)χ(y′′) + χ_{−}(y′)χ(xy′′))

+ ∑_{(x)(y)} χ_{−}(x′y′)χ(x′′y′′)).

These two expressions are easily seen to be equal using the commutativity of the algebra A, the character property for χ and the induction hypothesis.

•

Remark : define the Bogoliubov character as the map b : G → L(H, A) recursively given by :

b(φ)(x) = φ(x) + ∑_{(x)} φ_{−}(x′)φ(x′′).

Then the components of φ in the birkhoff decomposition read :

φ_{−} = −π ∘ b(φ), φ_{+} = (I − π) ∘ b(φ).

II.6. The BCH approach to Birkhoff decomposition

Let 𝔤 a Lie algebra endowed with a decreasing filtration :

𝔤 = 𝔤_{1} ⊃ 𝔤_{2} ⊃ ⋯ ⊃ 𝔤_{n} ⊃ ⋯

such that the intersection of the 𝔤_{i}’s is reduced to {0}. We ask for the inclusion :

[𝔤_{i}, 𝔤_{j}] ⊂ 𝔤_{i+j},

so in particular 𝔤 is a pro-nilpotent Lie algebra. Suppose that 𝔤 is complete for the (metric) topology defined by this filtration. The Baker-Campbell-Hausdorff series defines then a pro-nilpotent group law on 𝔤 :

X.Y = X + Y + \frac{1}{2}[X, Y] + \frac{1}{12}([X, [X, Y]] + [Y, [Y, X]]) + ⋯

The following proposition is implicitly used in [EGK2] :

Proposition II.6.1. For any linear map R : 𝔤 → 𝔤 preserving the filtration there exists a (usually non-linear) map χ_{R} : 𝔤 → 𝔤 such that (χ_{R} − Id_{𝔤})(𝔤_{i}) ⊂ 𝔤_{2i} for any i ≥ 1, and such that, with R̃ := Id_{𝔤} −R we have :

∀X ∈ 𝔤, X = R(χ_{R}(X)).R̃(χ_{R}(X)).  (∗)

Proof. Let us introduce for any X, Y ∈ 𝔤 the following expression :

δ(X, Y) = X.Y − X − Y = \frac{1}{2}[X, Y] + \frac{1}{12}([X, [X, Y]] + [Y, [Y, X]]) + ⋯

Then equation (*) can be rewritten as :

χ_{R}(X) = F_{X}(χ_{R}(X)),

with F_{X} : 𝔤 → 𝔤 defined by :

F_{X}(Y) = X − δ(R(Y), R̃(Y)).

This map F_{X} is a contraction with respect to the metric associated with the filtration : indeed if Y, ε ∈ 𝔤 with ε ∈ 𝔤_{n}, we have :

F_{X}(Y + ε) − F_{X}(Y) = δ(R(Y), R̃(Y)) − δ(R(Y + ε), R̃(Y + ε)).

Right-hand side is a sum of iterated commutators in each of which ε does appear at least once. So it belongs to 𝔤_{n+1}. So the sequence F_{X}^{n}(Y) converges in 𝔤 to a unique fixed point χ_{R}(X) for F_{X}.

Let us remark that for any X ∈ 𝔤_{i}, the element F_{X}(X) − X belongs to 𝔤_{2i}. Now taking X as starting point it is obvious from the expression :

χ_{R}(X) − X = ∑_{k=1}^{+∞}(F_{X}^{k} − F_{X}^{k−1})(X)

that for any X ∈ 𝔤_{i}, the element χ_{R}(X) − X belongs to 𝔤_{2i}.

•

Let H be a connected filtered Hopf algebra, let A a commutative algebra endowed with a splitting A = A_{−} ⊕ A_{+} as in § II.5, and let π be the projection on A_{−} parallel to A_{+}. Now take for 𝔤 any of the Lie algebras 𝔤, 𝔤_{1}, 𝔤_{2} of § II.3 and II.4, set G = exp_{∗}(𝔤) and set R(X) = π ∘ X. Of course R makes sense on L(H, A). Equation (*) then yields the following equality in the group G, G_{1} or G_{2} :

e^{∗X} = e^{∗R(χ_{R}(X))} ∗ e^{∗_{R}(χ_{R}(X))},  (∗∗)

as map R respects the decreasing filtration introduced in § II.3. Now, thanks to Rota-Baxter relation in A :

π(a)π(b) = π(π(a)b + π(b)a − ab),

K. Ebrahimi-Fard, L. Guo and D. Kreimer derive in [EGK2] two identities involving the Bogoliubov character :

e^{∗−R(χ_{R}(X))} = R(b(e^{∗X})), e^{∗_{R}(χ_{R}(X))} = −R̃(b(e^{∗X})).

So (**) is indeed the Birkhoff decomposition of the element φ = e^{∗X} of G, G_{1} or G_{2}, namely :

φ_{−} = e^{∗−R(χ_{R}(X))}, φ_{+} = e^{∗_{R}(χ_{R}(X))}.

Of course when H is cocommutative the convolution product is commutative, the Lie algebras involved are abelian and the situation simplifies greatly, as χ_{R} = Id_{𝔤} for any linear map R here [EGK1]. Further developments on general Rota-Baxter algebras can be found in [EGK2] and [EGK3]. Remark : Rota-Baxter identity for R just guarantees that equation (**) gives a Birkhoff decomposition. When R is idempotent (which is indeed the case in the setting of § II.5, where R is built up from projection π) this decomposition is unique and is given either by (**) or by the recursive formulas of Theorem II.5.1. On the other hand, with an idempotent R which does not verify Rota-Baxter identity (or, which is the same, with a direct sum decomposition of the algebra A on two components A_{+} and A_{−} which are not subalgebras of A), we still get a unique Birkhoff decomposition along the lines of theorem II.5.1 for the groups G and G_{2}, which does not coincide with (**). But Rota-Baxter identity arises in an essential way, as we have seen, in order to get the Birkhoff decomposition for the groups G_{1} of A-valued characters.

II.7. Renormalized traces and characters

Keeping the same notations we take k = ℂ as ground field and we specialize to the case when A is the field of meromorphic functions (resp. the field of germs of meromorphic functions at z_{0}), A_{+} is the algebra of meromorphic functions which are holomorphic at z_{0} (resp. the algebra of germs of holomorphic functions at z_{0}) and A_{−} is the non-unital algebra (z − z_{0})^{−1}ℂ[(z − z_{0})^{−1}]. Applying the projection π to such a meromorphic fuction amounts to “take its divergent part at z_{0}”. This particular splitting is called the minimal subtraction scheme. It is by no means unique : for example for any automorphism θ of the field of (germs of) meromorphic functions, we can consider the splitting :

A = A_{+}^{θ} ⊕ A_{−}^{θ},

with A_{±}^{θ} = θ(A_{±}). The corresponding projection on A_{−}^{θ} is given by π^{θ} = θ ∘ π ∘ θ^{−1}. As an example of automorphism θ fix a constant c, consider the change of variable z ↦ z′ such that :

\frac{1}{z′ − z_{0}} = \frac{1}{z − z_{0}} + c

(hence z′ = z_{0} + \frac{z − z_{0}}{1 + c(z − z_{0})}), and set θ(f)(z) = f(z′).

Considering a linear map φ from H to A (and a particular splitting of A) we can consider its Birkhoff decomposition φ = φ_{−}^{∗−1} ∗ φ_{+} given by Theorem II.5.1, and evaluate φ_{+}(x) at z = z_{0} for any x ∈ H. This gives a linear map φ_{+}^{z_{0}} from H to ℂ which we call the renormalized value of φ at z = z_{0}. According to Theorem II.5.1 the renormalized value of a cocycle at z_{0} is a trace, and the renormalized value of a character at z = z_{0} is a ℂ-valued character. All this procedure depends in an essential way on the choice of the renormalization scheme, i.e. the splitting of A.

II.8. More on connected graded Hopf algebras

Let H be a connected graded Hopf algebra. The grading induces a biderivation Y defined on homogeneous elements by :

\begin{matrix}Y : H_{n} & ⟶ H_{n} \\ x & ⟼ nx. \end{matrix}

Exponentiating we get a one-parameter group θ_{t} of automorphisms of the Hopf algebra H, defined on H_{n} by :

θ_{t}(x) = e^{nt}x.

Lemma II.8.1. φ ↦ φ∘Y is a derivation of (L(H, A), ∗), and φ ↦ φ∘θ_{t} is an automorphism of (L(H, A), ∗) for any complex t.

Proof. We compute for φ, ψ ∈ L(H, A) and for an homogeneous element x of H :

(φ ∗ ψ) ∘ Y (x) = |x| ∑_{(x)} φ(x_{1})ψ(x_{2})

= ∑_{(x)}(|x_{1}| + |x_{2}|)φ(x_{1})ψ(x_{2})

= ∑_{(x)}(φ ∘ Y)(x_{1})ψ(x_{2}) + φ(x_{1})(ψ ∘ Y)(x_{2})

= ((φ ∘ Y) ∗ ψ + φ ∗ (ψ ∘ Y))(x),

which shows the first assertion. The second part of the proposition is proven similarly.

•

Using the fact that e ∘ Y = 0 we easily compute for any derivation α with values in A :

(α ∘ Y)(xy) = α(Y (x).y + x.Y (y))
= (α ∘ Y)(x)e(y) + (e ∘ Y)(x)α(y) + α(x)(e ∘ Y)(y) + e(x)(α ∘ Y)(y)
= (α ∘ Y)(x)e(y) + e(x)(α ∘ Y)(y).

So we have proved :

Lemma II.8.2. The map α ↦ α ∘ Y is a linear automorphism of the space of derivations of H with values in A. Its inverse is given by α ↦ α ∘ Y^{−1}, where Y^{−1}(x) = |x|^{−1}x for x homogeneous of positive degree, and Y^{−1}(1) = 0.

Remark : the notation Y^{−1} is of course slightly incorrect, as the inverse of Y does not make sense on H_{0}. The convention Y^{−1}(1) = 0 is arbitrary : any other value of Y^{−1}(1) would give the same result, as derivations with values in A vanish at 1.

II.9. Examples

II.9.1. The Hopf algebra of positive integers

This example is a simplified version of the one given by D. Kreimer in [K2 § 2.1]. Consider the algebra N of the multiplicative semigroup ℕ^{∗} = {1, 2, 3, …} of positive integers. As a vector space it admits a basis (e_{n})_{n∈ℕ^{∗}} with product given by e_{n}.e_{m} = e_{nm} and extended by linearity. We endow N with a structure of commutative cocommutative connected graded Hopf algebra thanks to the decomposition of any integer into a product of prime factors : namely we set Δ(e_{1}) = e_{1} ⊗ e_{1}, and for any prime p :

Δ(e_{p}) = e_{p} ⊗ e_{1} + e_{1} ⊗ e_{p},

and we extend Δ to an algebra isomorphism. Hence,

Δ(e_{p_{1}···p_{k}}) = ∑_{I⨿J={1,...,k}} e_{p_{I}} ⊗ e_{p_{J}} ,

where p_{I} denotes the product of the primes p_{j}, j ∈ I. The grading is clearly given by the number of prime factors (including multiplicities). The antipode is given by :

S(e_{n}) = (−1)^{|n|}e_{n}.

Suppose that the ground field is k = ℂ. The map n ↦ n^{z} defines a character φ of N with values into the holomorphic functions. Then the Riemann Zeta function is nothing but the evaluation of φ on the element :

ω = e_{1} + e_{2} + e_{3} + ⋯ = ∏_{p prime} \frac{1}{1 − e_{p}} .

Here 1/(1 − e_{p}) stands for the infinite sum : e_{1} + e_{p} + e_{p^{2}} + · · ·. Of course ω is not an element of N : it makes sense (as well as the abstract Euler product expansion on the right-hand side) only in the completion of N with respect to the fine filtration defined by the vector space grading d(n) = n − 1. But evaluating the character φ on both sides of this equality gives the well-known Euler product expression of the Zeta function.

II.9.2. Tensor and symmetric algebras

The tensor Hopf algebra T (V) of any vector space V (cf. Example I.6.2) is obviously graded. The symmetric Hopf algebra is a particular case of enveloping Hopf algebra, with V viewed as an abelian Lie algebra. The Hopf algebra S(V) is a cocommutative commutative connected graded Hopf algebra. Note that an enveloping algebra is not graded in general, since the quotienting ideal generated by x ⊗ y − y ⊗ x − [x, y] is not homogeneous.

II.9.3. Planar decorated rooted trees

We borrow in this section some material from [F]. A planar rooted tree is an oriented connected contractible graph, with a finite number of vertices, together with an embedding of it into the plane, such that only one vertex has only outgoing edges (the root). We have drawn below the planar rooted trees with four vertices :

Figure 1 : the planar rooted trees with four vertices.

Let T be the set of planar rooted trees. Let V be a vector space on some field k, and let t be a planar rooted tree. The space of decorations of t by V is the vector space V^{⊗t}. A planar rooted tree t together with an indecomposable element of V^{⊗t} is called a decorated rooted tree. Let us consider the vector space :

T_{V} = ⨁_{t∈T} V^{⊗t},

let H_{V} be the (noncommutative) free algebra generated by T_{V}. Products of decorated trees (decorated forests) generate H_{V} as a graded vector space, the degree of a decorated forest being given by the total number of vertices. The connected graded Hopf algebra structure on H_{V} is given by the co-unit ε sending 1 to 1 and any nonempty decorated forest to 0, and by a coproduct which we describe shortly here :

An elementary cut on a tree is a cut on some edge of the given tree. An admissible cut is a cut such that any path starting from the root contains at most one elementary cut. The empty cut is considered as elementary, as well as the total cut, i.e. a cut below the root. A cut on a forest is said to be admissible if its restriction to any tree factor is admissible. Any elementary cut c sends a forest F to a couple (P^{c}(F), R^{c}(F)), the crown and the trunk respectively. The trunk of a tree is a tree, but the crown of a tree is a forest. Let Adm(F) the set of admissible cuts of the forest F, and let Adm^{∗}(F) the set of elementary cuts discarding the empty cut and the total cut. The coproduct :

Δ(F) = ∑_{c∈Adm F} P^{c}(F) ⊗ R^{c}(F)

is graded, co-associative and compatible with the product [F]. The compatibility with the product is clear (due to the definition of an admissible cut for a forest). There is a beautiful proof of the co-associativity in [F] using induction on the degree and grafting of any forest on a decorated root. We propose here a more intuitive proof : say that a couple (c_{1}, c_{2}) of cuts is bi-admissible if both cuts c_{1}, c_{2} are admissible and if c_{1} never bypasses c_{2}, i.e. if c_{2} never cuts the trunk of c_{1}. Any bi-admissible couple c = (c_{1}, c_{2}) of cuts c defines a crown P^{c}(F) = P^{c_{2}}(F), a trunk R^{c}(F) = R^{c_{1}}(F), and a middle M^{c}(F) :

Figure 2 : an example of bi-admissible couple of cuts. From thickest to thinnest : trunk, middle and crown.

Let Adm_{2} F the set of bi-admissible couples of cuts of the forest F. It is quite straightforward to set down the formula for the iterated coproduct :

(Δ ⊗ I) ∘ Δ(F) = (I ⊗ Δ) ∘ Δ(F) = ∑_{c∈Adm_{2} F} P^{c}(F) ⊗ M^{c}(F) ⊗ R^{c}(F).

Of course the n-fold iterated coproduct admits a similar expression, involving n-admissible n-uples of admissible cuts and n + 1 “level segments” of the forest, from the crown down to the trunk.

By corollary II.2.2 the connected graded bialgebra H_{V} thus obtained admits an antipode given on Ker ε by any of the two recursive formulas :

\begin{aligned}S(F) & = −F − ∑_{c∈Adm^{∗}(F)} S(P^{c}(F)).R^{c}(F) \\ & = −F − ∑_{c∈Adm^{∗}(F)} P^{c}(F).S(R^{c}(F)). \end{aligned}

The square of the antipode does not in general coincide with the identity.

II.9.3. Decorated rooted trees

The construction is the same except that we consider rooted trees independently from any embedding into the plane, and we consider the free commutative algebra generated by decorated rooted trees. We thus obtain a commutative Hopf algebra H′_{V} which is clearly a quotient of H_{V}. This Hopf algebra is thoroughly investigated in [F].

III. Hopf algebras of Feynman graphs

We treat this example (more exactly this family of examples) in a separate section for two main reasons : firstly the Hopf algebras appearing there are pointed but not connected, and secondly this is the very example where a link is established with quantum field theory.

The non-connectedness is not a very serious problem : as we shall see we can reason on a connected quotient and go back. The formula for the coproduct will differ slightly from that of Connes-Kreimer in order to deal with this non-connectedness problem, but both will agree on the connected quotient. We follow [K1] quite closely, with some modifications in order to allow self-loops.

III.1. Discarding exterior structures

Anyone a little bit familiar with quantum field theory knows that Feynman graphs are made of internal and external edges of different types, and that an external edge comes with a vector attached to it (an exterior momentum). The sum of all exterior momenta of a given graph must be equal to zero, reflecting the global conservation of momenta in an interaction. The Feynman rules attach to a graph together with such an external structure an integral which can be divergent. This integral can be regularized by various procedures, among them dimensional regularization : the idea is to “let the dimension of the space of momenta vary in the complex numbers”, a procedure which has been recently given a precise geometrical contents by A. Connes and M. Marcolli ([CM2] § 15). The divergent integral is now replaced by a meromorphic function with poles at least at the entire dimensions where the original integral diverges [C Chap. 4], [E].

The approach of renormalization by A. Connes and D. Kreimer can be summarized as follows : organize Feynman graphs with their exterior structures into a graded Hopf algebra, understand the (regularized, e.g. by means of dimensional regularization) Feynman rules as a character of this Hopf algebra with values into some algebra A (e.g. the meromorphic functions), choose a renormalization scheme, i.e. a splitting A = A_{+} ⊕ A_{−} into two subalgebras, apply the method of § II.5 and II.6 to extract a renormalized value, and finally recognize that this method agrees with algorithms already developed by physicists, such as the Bogoliubov-Parasiuk-Hepp-Zimmerman (BPHZ) algorithm.

Our first step will consist in constructing a Hopf algebra from Feynman diagrams without exterior structure (i.e. with exterior momenta nullified).

III.2. Operations on Feynman graphs

A Feynman graph is a (non-oriented, non-planar) graph with a finite number of vertices and edges. An internal edge is an edge connected at both ends to a vertex (which can be the same in case of a self-loop), an external edge is an edge with one open end, the other end being connected to a vertex.

A Feynman graph is called by physicists vacuum graph, tadpole graph, self-energy graph, resp. interaction graph if its number of external edges is 0,1, 2, resp. > 2.

The edges (internal or external) will be of different types labelled by a positive integer (1, 2, 3, …), each type being represented by the way the corresponding edge is drawn (full, dashed, wavy, various colours, etc...). Let τ(e) ∈ ℕ^{∗} be the type of the edge e. For any vertex v let st(v) be the star of v, i.e. the set of all edges attached to v, with self-loops counted twice. Hence the valence of the vertex is given by the cardinal of st(v). Finally to each vertex we associate its type, the sequence (n_{1}, …, n_{r}) of positive integers where n_{j} stands for the number of edges of type j in st(v). Let T (v) be the type of the vertex v

For example, in the φ^{n} theory there is only one type of edge, and two types of vertices : the bivalent vertices and the n-valent vertices. In quantum electrodynamics there are two types of edges : the fermion edges (usually drawn full), and the boson edges (usually drawn wavy), and three types of vertices : bivalent boson-boson vertices, bivalent fermion-fermion vertices, and trivalent vertices with two fermion edges and one boson edge. Most of the pictures will be drawn in φ^{3} or φ^{4} theory, or in quantum electrodynamics.

A one-particle irreducible graph (in short, 1PI graph) is a connected graph which remains connected when we cut any internal edge. A disconnected graph is said to be locally 1PI if any of its connected components is 1PI. The residue of a connected graph is the graph with only one vertex obtained by shrinking all internal edges to a point. Of course any connected graph has the same type as its residue.

Figure 3 : A QED interaction graph and its residue.

A subgraph of a Feynman graph is either the empty graph, or a nonempty (connected or disconnected) set of internal edges together with the vertices they encounter and the stars of those vertices. A proper subgraph of Γ is a subgraph different from the empty graph or the whole graph Γ itself. If γ is a subgraph inside a graph Γ, the contracted graph Γ/γ is the graph obtained by replacing all connected components of γ by their residues inside Γ. As an example the residue of a graph Γ is equal to Γ/Γ.

Figure 4 : A subgraph γ inside a graph Γ in φ^{3} theory. The contracted graph Γ/γ does not belong to φ^{3}.

III.3. The graded Hopf algebra structure

Fix a set T = {T_{1}, …, T_{k}} of finite sequences of positive integers, which will be the possible vertex types we want to deal with. Let V_{T} be the vector space generated by all connected 1PI Feynman graphs with vertex types in T , and all residues of those. Let B_{T} = S(V_{T}) be the free commutative algebra generated by V. We shall identify the unit 1 with the empty graph and any element of B_{T} with a linear combination of disconnected locally 1PI graphs. The algebra structure is obvious, the co-unity is given by ε(1) = 1 and ε(Γ) = 0 for any nonempty graph Γ. The grading (at least, one possible grading) is given on connected graphs by the loop number :

L := I − V + 1,

where I is the number of internal edges and V is the number of vertices of a given graph. This grading is extended to non-connected graphs in such a way that it is compatible with the algebra structure. It is important to notice that any nonempty subgraph has a nonvanishing loop number. The coproduct is given on connected 1PI graphs by the following formula :

Δ(Γ) = ∑_{γ subgraph of Γ Γ/γ∈V_{T}} γ ⊗ Γ/γ

= Γ ⊗ res Γ + 1 ⊗ Γ + ∑_{γ proper subgraph of Γ Γ/γ∈V_{T}} γ ⊗ Γ/γ  if L(Γ) ≥ 1,

Δ(Γ) = Γ ⊗ Γ  if L(Γ) = 0,

and extended to non-connected graphs by multiplicativity. We leave it to the reader as an easy exercice to show that the coproduct respects the loop number as well. Figure 5 below illustrates a coproduct computation in φ^{3} theory. Two terms of the sum have been removed because the corresponding contracted graphs have a vertex the type of which is outside T (here a pentavalent and an hexavalent vertex respectively), and then does not belong to V_{T}. On the other hand residues with any number of external edges are allowed.

Figure 5 : an example of coproduct in φ^{3} theory.

Figure 6 below illustrates another coproduct computation in φ^{3} theory, with a bivalent vertex arising in the contracted graph :

Figure 6 : another example of coproduct in φ^{3} theory.

Proposition III.3.1. B_{T} is a pointed graded bialgebra.

Proof. All axioms of a pointed graded bialgebra have been already given by the construction, except coassociativity of the coproduct. But we have for any 1PI graph of positive degree :

(Δ ⊗ I)Δ(Γ) = ∑_{{δ⊂γ⊂Γ \atop γ/δ∈B_{T} , Γ/γ∈B_{T}}} δ ⊗ γ/δ ⊗ Γ/γ,

whereas :

(I ⊗ Δ)Δ(Γ) = ∑_{{δ⊂Γ,_{γ}⊂Γ/δ \atop Γ/δ∈B_{T} , (Γ/δ)/_{γ}∈B_{T}}} δ ⊗ γ̃ ⊗ (Γ/δ)/γ̃.

There is an obvious bijection γ ↦ γ̃ = γ/δ from subgraphs of Γ containing δ onto subgraphs of Γ/δ, given by shrinking δ. As we have the obvious “transitive shrinking property” :

Γ/γ = (Γ/δ)/γ̃,

the two expressions coincide.

•

In order to build up a graded Hopf algebra from B_{T}, two choices are possible : first we can add formally the inverses of the grouplike elements, i.e. the degree zero graphs : let Σ be the set of degree zero connected 1PI graphs, let Σ^{−1} be another copy of the same set, with elements labelled γ^{−1}, γ ∈ Σ. Let Ṽ_{T} be the vector space generated by V_{T} and Σ^{−1}, and consider :

H̃_{T} = S(Ṽ_{T})/J,

where J is the ideal generated by γγ^{−1} − 1, γ ∈ Σ. The coproduct on S(V_{T}) is extended to S(Ṽ_{T}) by saying that the elements of Σ^{−1} are grouplike. The ideal J is the a bi-ideal, and so H̃_{T} is a pointed graded bialgebra. An antipode is easily given inductively with respect to the degree, as any degree zero element has an antipode given by S(γ) = γ^{−1}, S(γ^{−1}) = γ for any γ ∈ Σ.

The second option consists in killing the degree zero graphs (except the empty graph). We set :

H_{T} = B_{T} /K,

where K is the ideal generated by γ − 1, γ ∈ Σ. It is easily seen to be a bi-ideal. The quotient is then a connected graded bialgebra, hence a Hopf algebra thanks to corollary II.3.2. We can identify the quotient with S(V′_{T}), where V′_{T} stands for the vector space generated by connected 1P I graphs with loop number ≥ 1. The coproduct is then given by Kreimer’s formula :

Δ(Γ) = Γ ⊗ 1 + 1 ⊗ Γ + ∑_{γ proper subgraph of Γ Γ/γ∈V_{T}} γ ⊗ Γ/γ.

III.4. External structures

We shall be very sketchy here. Let W be a finite-dimensional vector space (the momentum space). Keeping the notations of § III.3 and following [CK 2] and [K 1], a specified graph will be a couple (Γ, σ) where Γ is a connected graph in V_{T} with E external lines, and σ is a distribution on the vector subspace M_{Γ} = M_{E} ⊂ W^{E} defined by :

M_{E} = {(p_{1}, …, p_{E}), ∑_{k=1}^{E} p_{k} = 0}.

In order to get a Hopf algebra structure for specified graphs we must discriminate further the type of a vertex : once the number of edges of each type is fixed for a vertex, we add an extra nonzero natural number, so that there are “several kinds of vertices of the same type”. This comes from the lagrangian of the given quantum field theory we are dealing with : each monomial of degree n_{i} with respect to the field ϕ_{i} (i ∈ {1, …, k}) gives rise to vertices of type T = (n_{1}, …, n_{k}), and there are as many kinds of vertices of type T as monomials of “field degree” T inside the lagrangian. For example, in φ^{3} theory with mass, terms (m^{2}/2)φ^{2} and (∂φ)^{2}/2 give rise to two different kinds of bivalent vertices. When taking residues we must specify the kind for the unique remaining vertex : then when considering a contracted graph Γ/γ we must consider the kind of every contracted vertex (corresponding to a connected component of the subgraph γ). This gives rise to contracted graphs Γ/γ(i) where i is a multi-index.

To any vertex of kind (T, i) corresponds a specific distribution σ_{T,i} on M_{Γ}, where Γ is any graph whose residue gives a vertex of type T. This extends to non-connected graphs by considering multi-indices i. Now T stands for the set of all kinds (T, i) of vertices we can encounter, V_{T} stands for the space generated by all connected 1PI graphs with vertex kinds in T , and all residues of those. Let V′_{T} the space generated by all connected 1PI Feynman graphs with vertex kinds in T and nonzero loop number, let (V′_{T})_{E} the subspace of V′_{T} of graphs with E external edges, and finally let W′_{T} the corresponding space of specified graphs :

W′_{T} = ∑_{E=0}^{∞}(V′_{T})_{E} ⊗ D′(M_{E}).

We directly give the connected version of the Hopf algebra : it is given by H_{T} = S(W′_{T}), and the coproduct is given on connected specified graphs by :

Δ(Γ, σ) = (Γ, σ) ⊗ 1 + 1 ⊗ (Γ, σ) + ∑_{γ proper subgraph of Γ} ∑_{i, Γ/γ(i)∈V_{T}} (γ, σ_{T,i}) ⊗ (Γ/γ(i), σ).

IV. An approach to the renormalization group

Keeping the notations of paragraph II, we denote by H a connected graded Hopf algebra, and by A the algebra of germs of meromorphic functions at some z_{0} ∈ ℂ. The algebra A admits a splitting into two subalgebras :

A = A_{+} ⊕ A_{−},

where A_{+} is the algebra of germs of holomorphic functions at z_{0}, and A_{−} = (z−z_{0})^{−1}ℂ[(z− z_{0})^{−1}]. We denote by Y (resp. θ_{t}) the biderivation (resp. the one-parameter group of automorphisms) of the Hopf algebra H induced by the graduation (cf. § II.6). As in paragraph II we denote by G the group of the elements φ ∈ L(H, A) such that φ(1) = 1_{A} (with the convolution products), and by 𝔤 the subalgebra of L(H, A) of the elements φ ∈ L(H, A) such that φ(1) = 0.

Recall that G = exp 𝔤. As in § II we shall consider the subgroups G_{1} (resp. G_{2}) of G formed by the characters of H with values in A (resp. by the elements of G which enjoy the cocycle property), as well as the Lie subalgebras 𝔤_{1} (resp. 𝔤_{2}) of derivations of H with values in A (resp. of 𝔤 which enjoy the cocycle property). We have G_{1} = exp 𝔤_{1} and G_{2} = exp 𝔤_{2}.

IV.1. The renormalization map

We settle here a bijection R : 𝔤 → 𝔤 thanks to the biderivation Y :

Proposition IV.1.1. The equation :

φ ∘ Y = φ ∗ γ  (E)

defines a bijective correspondence :

\begin{matrix}R̃ : G & ⟶ 𝔤 \\ φ & ⟼ γ. \end{matrix}

Equivalently the equation :

e^{∗α} ∘ Y = e^{∗α} ∗ γ  (E′)

defines a (non-linear) bijective correspondence :

\begin{matrix}R : 𝔤 & ⟶ 𝔤 \\ α & ⟼ γ, \end{matrix}

and R = R̃ ∘ exp.

Proof. Equation (E) yields for any homogeneous x ∈ H :

|x|φ(x) = γ(x) + ∑_{(x)} φ(x′)γ(x′′),

which determines γ (recursively in |x|) from φ and vice-versa, starting from φ(1) = 1_{A} and γ(1) = 0. In other words equation (E) determines a bijection R̃ from G to 𝔤 such that γ = R̃(φ). The remainder of prop. IV.1.1 follows then immediately.

•

Equation (E′) yields the following explicit expression for R :

R(α) = e^{∗−α} ∗ (e^{∗α} ∘ Y).

There is another explicit formula :

Proposition IV.1.2.

R(α) = ∫_{0}^{1} e^{∗−sα} ∗ (α ∘ Y) ∗ e^{∗sα} ds.

Proof. for any u ∈ ℝ we have :

e^{∗uα} ∘ Y = e^{∗uα} ∗ R(uα).

Setting u = t + s and using the group property e^{∗(t+s)}α = e^{∗tα} ∗ e^{∗sα} as well as the derivation property :

(e^{∗tα} ∗ e^{∗sα}) ∘ Y = (e^{∗tα} ∘ Y) ∗ e^{∗sα} + e^{∗tα} ∗ (e^{∗sα} ∘ Y),

we get :

e^{∗(t+s)α} ∘ Y = e^{∗(t+s)α} ∗ (R(sα) + e^{∗−sα} ∗ R(tα) ∗ e^{∗sα}).

Set γ(t) = R(tα) : the above equation reads :

γ(t + s) = γ(s) + e^{∗−sα} ∗ γ(t) ∗ e^{∗sα}.

We have γ(0) = 0, and differentiating this equation with respect to s at s = 0 yields :

γ̇(t) = γ̇(0) + [γ(t), α].

Differentiating once again with respect to t gives then :

γ̈(t) = [γ̇(t), α].

The solution of this first order differential equation is given by :

γ̇(t) = e^{∗−tα} ∗ γ̇(0) ∗ e^{∗tα}.

Expanding the equation e^{∗tα} ∘ Y = e^{∗tα} ∗ γ(t) up to order 1 in t = 0 yields immediately :

γ̇(0) = α ∘ Y.

Integrating and setting t = 1 establishes then proposition IV.1.2.

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Corollary IV.1.3. Correspondence R sends A-valued derivations to A-valued derivations and cocycles to cocycles.

Proof. First assertion follows immediately from propositions IV.1.2, II.4.2 and II.8.2. Second assertion follows directly from proposition IV.1.2.

•

Remark : If the Hopf algebra H is cocommutative, then thanks to the commutativity of A, the convolution product is commutative. The correspondence R becomes then linear and we simply have :

R(α) = α ∘ Y.

IV.2. Inverting R̃ : the scattering map

We shall give an explicit expression of the map R̃^{−1} : 𝔤 → G. It takes the form :

R̃^{−1}(γ) = lim_{t→+∞} exp −tA exp tB,

(cf. theorem IV.2.1 below), where A and B live in a semi-direct product Lie algebra 𝔤̃ = 𝔤 ×^{|} ℂ. We have to describe this semi-direct product and the corresponding semidirect product group G̃ = G ×^{|} ℂ, and then we must endow G̃ with a topology so that the above limit makes sense. We adapt here the proof of Theorem 2 in [CK2]. To be precise, we define the Lie algebra :

g̃ = 𝔤 ×^{|} ℂ.Z_{0},

where the action of Z_{0} on 𝔤 is given by the derivation :

Z_{0}(γ) = γ ∘ Y

(see lemma II.7.1). The corresponding group is G̃ = G ×^{|} ℂ, where the right action of ℂ on G is given by :

φ.t = φ ∘ θ_{t},

so that the product is given by (φ, t)(ψ, s) = (φ ∗ (ψ ∘ θ_{t}), t + s). We shall not dig out a Lie group structure for G̃ here, but we shall define the exponential map exp : 𝔤̃ → G̃. It should of course coincide with the exponential already defined on G, and should verify :

exp tZ_{0} = (e, t)

so that exp tZ_{0} indeed acts on G by composition with θ_{t} = exp tY on the right. We should be able in principle to define exp(tZ_{0} + γ) by means of the Baker-Campbell-Hausdorff formula as long as convergence problems can be handled here. We prefer, like in [CK2], to give an alternative definition based on Araki’s expansion formula [Ar] :

exp(tZ_{0} + γ) = ∑_{n=0}^{∞} ∫_{∑_{j=0}^{n} u_{j}=1, u_{j}≥0} exp(u_{0}tZ_{0})γ exp(u_{1}tZ_{0})γ ⋯ γ exp(u_{n}tZ_{0}) du_{1} ⋯ du_{n}.

Let us check that the sum above makes sense in our particular context : setting v_{j} = u_{j} + u_{j+1} + ⋯ + u_{n} we get :

exp(−tZ_{0}) exp(tZ_{0} + γ) = exp(−tZ_{0}).

exp(tZ_{0}). ∑_{n=0}^{∞} ∫_{0≤v_{n}≤···≤v_{1}≤1} exp(−tv_{1}Z_{0})γ exp(tv_{1}Z_{0}) ⋯ exp(−tv_{n}Z_{0})γ exp(tv_{n}Z_{0}) dv_{1} ⋯ dv_{n}

= ∑_{n=0}^{∞} ∫_{0≤v_{n}≤···≤v_{1}≤1} (γ ∘ θ_{−tv_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−tv_{n}}) dv_{1} ⋯ dv_{n}.

The sum here is well defined as a locally finite sum, as it ends up at n = n_{0} when evaluated at any x = H^{n_{0}}. It remains to check that the exponential thus defined enjoys the oneparameter group property. Indeed, for any s, t real we have :

exp t(Z_{0} + γ) exp s(Z_{0} + γ) = e^{tZ_{0}}(∑_{p=0}^{∞} t^{p} ∫_{0≤v_{p}≤···≤v_{1}≤1} (γ ∘ θ_{−tv_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−tv_{p}}) dv_{1} ⋯ dv_{p}).

e^{sZ_{0}}(∑_{p=0}^{∞} s^{q} ∫_{0≤w_{q}≤···≤w_{1}≤1} (γ ∘ θ_{−sw_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−sw_{q}}) dw_{1} ⋯ dw_{q})

= e^{(t+s)Z_{0}} ∑_{p,q=0}^{∞} t^{p}s^{q} ∫ ∫_{0≤v_{p}≤···≤v_{1}≤1, 0≤w_{q}≤···≤w_{1}≤1}

(γ ∘ θ_{−s−tv_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−s−tv_{p}}) ∗ (γ ∘ θ_{−sw_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−sw_{q}}) dv_{1} ⋯ dv_{p}dw_{1} ⋯ dw_{q}

= e^{(t+s)Z_{0}} ∑_{p,q=0}^{∞} ∫ ∫_{0≤v_{p}≤···≤v_{1}≤t, 0≤w_{q}≤···≤w_{1}≤s}

(γ ∘ θ_{−s−v_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−s−v_{p}}) ∗ (γ ∘ θ_{−w_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−w_{q}}) dv_{1} ⋯ dv_{p}dw_{1} ⋯ dw_{q}

= e^{(t+s)Z_{0}} ∑_{n=0}^{∞} ∑_{p+q=n} ∫ ∫_{s≤v_{p}≤···≤v_{1}≤t+s, 0≤w_{q}≤···≤w_{1}≤s}

(γ ∘ θ_{−v_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−v_{p}}) ∗ (γ ∘ θ_{−w_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−w_{q}}) dv_{1} ⋯ dv_{p}dw_{1} ⋯ dw_{q}

= e^{(t+s)Z_{0}} ∑_{n=0}^{∞} ∫_{s≤u_{p}≤···≤u_{1}≤t+s}

(γ ∘ θ_{−u_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−u_{n}}) du_{1} ⋯ du_{n} = exp(t + s)(Z_{0} + γ).

We can now state the main theorem of this section :

Theorem IV.2.1. Let γ ∈ 𝔤. Then :

1) For any real t the product exp −tZ_{0} exp t(Z_{0} + γ) belongs to G.

2) The product above admits a limit when t → +∞ for the topology on G induced by the simple convergence topology on L(H, A).

3) The inverse of the renormalization map is given by :

R̃^{−1}(γ) = lim_{t→+∞} exp −tZ_{0} exp t(Z_{0} + γ).

4) R̃^{−1} sends 𝔤_{1} into G_{1} and 𝔤_{2} into G_{2}.

Proof. The first assertion comes directly from the expression :

exp(−tZ_{0}) exp(tZ_{0} + tγ) = ∑_{n=0}^{∞} ∫_{0≤v_{n}≤···≤v_{1}≤1} (tγ ∘ θ_{−tv_{1}}) ∗ ⋯ ∗ (tγ ∘ θ_{−tv_{n}}) dv_{1} ⋯ dv_{n}.

The right-hand side belongs manifestly to G. Change of variables v_{j} → tv_{j} yields :

exp(−tZ_{0}) exp(tZ_{0} + tγ) = ∑_{n=0}^{∞} ∫_{0≤v_{n}≤···≤v_{1}≤t} (γ ∘ θ_{−v_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−v_{n}}) dv_{1} ⋯ dv_{n}.

To prove the second assertion it suffices to prove that the integrals :

I_{n} := ∫_{0≤v_{n}≤···≤v_{1}≤+∞} (γ ∘ θ_{−v_{1}}) ∗ ⋯ ∗ (γ ∘ θ_{−v_{n}}) dv_{1} ⋯ dv_{n}

converge, as the sum I_{0} + I_{1} + I_{2} + ⋯ is locally finite. The convergence is easily seen by induction on n : indeed we have I_{0} = e and the crucial equality valid for any x ∈ Ker ε :

Y^{−1}(x) = ∫_{0}^{∞} θ_{−t}(x) dt.

It follows that we have for any a ∈ 𝔤 :

∫_{0}^{∞} a ∘ θ_{−t} dt = a ∘ Y^{−1}.

A simple computation then gives :

\begin{aligned}I_{n} & = ∫_{0}^{∞}(I_{n−1} ∗ γ) ∘ θ_{−v_{n}} dv_{n} \\ & = (I_{n−1} ∗ γ) ∘ Y^{−1}, \end{aligned}

which inductively shows the convergence of the integrals I_{n}. Now equation (E) can be rewritten as :

\begin{aligned}φ(x) & = (φ ∗ γ) ∘ Y^{−1}(x) ∀x ∈ Ker ε \\ φ(1) & = 1_{A}. \end{aligned}  (E′′)

As γ = R̃(φ) it means that :

R̃^{−1}(γ) = e + T (R̃^{−1}(γ)),

where T is the transformation of L = L(H, A) defined by :

\begin{aligned}T (ψ) & = (ψ ∗ γ) ∘ Y^{−1} \\ & = ∫_{0}^{∞}(ψ ∗ γ) ∘ θ_{−t} dt. \end{aligned}

Transformation T is a contraction on L for the distance associated with the filtration. R̃^{−1}(γ) is then the limit of the sequence (φ_{n}) defined by φ_{0} = e and φ_{n+1} = e + T (φ_{n}). A straightforward computation yields :

φ_{n} = ∑_{k=0}^{n} I_{k}.

Hence we have :

R̃^{−1}(γ) = ∑_{k=0}^{∞} I_{k},

which proves assertion 3). Finally assertion 4) comes from the fact that derivation Z_{0} acts on 𝔤_{1} and 𝔤_{2}. We can then consider semi-direct products :

\begin{matrix}𝔤̃_{1} = 𝔤_{1} ×^{|} ℂ.Z_{0},  G̃_{1} = G_{1} ×^{|} ℂ, \\ 𝔤̃_{2} = 𝔤_{2} ×^{|} ℂ.Z_{0},  G̃_{2} = G_{2} ×^{|} ℂ, \end{matrix}

and thus replace the group G by any of the two groups G_{1}, G_{2} in assertions 1), 2) and 3), which proves assertion 4) and ends the proof of the theorem.

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Corollary IV.2.2. The inverse of R : 𝔤 → 𝔤 is given by :

R^{−1}(γ) = lim_{t→+∞} Log(exp −tZ_{0} exp t(Z_{0} + γ)),

and R^{−1} sends 𝔤_{1} (resp. 𝔤_{2}) into 𝔤_{1} (resp. 𝔤_{2}).

IV.3. The residue

We keep the same notations as before except that we set z_{0} = 0 for notational simplicity. To any ψ ∈ L we associate a linear form Res ψ on H by extracting the z^{−1} term : more precisely if we have for any x ∈ H and for any z in some pointed neighbourhood of 0 :

ψ(x)(z) = ∑_{n=−N}^{+∞} ψ_{n}(x)z^{n}

with ψ_{n}(x) ∈ ℂ, then :

Res ψ(x) := ψ_{−1}(x).

IV.4. Renormalization map and Birkhoff decomposition

Following more closely A. Connes and D. Kreimer [CK2] we shall focus our attention on elements of G the Birkhoff decomposition of which shares an invariance property with respect to the action of the graduation. More precisely we define first a new action of ℂ on the group G by :

ψ^{t}(x)(z) = ψ(θ_{tz}(x))(z) = e^{tz|x|}ψ(x)(z).

We shall consider the elements ψ of G such that in the Birkhoff decomposition :

ψ^{t} = (ψ^{t})_{−}^{∗−1} ∗ (ψ^{t})_{+}

the polar part (ψ^{t})_{−} is independent of t. The motivation to look at this very specific property is that it is fulfilled by the characters of Hopf algebras of Feynman graphs obtained via Feynman rules and dimensional regularization. In physical terms this corresponds to the well-known fact that the counterterms do not depend on the choice of the arbitrary mass μ one must introduce in order to perform dimensional regularization. Let us say that ψ ∈ G enjoys property (Φ) if the above condition on the polar part of its Birkhoff decomposition is fulfilled. G^{Φ} will denote the subset of elements of G enjoying property (Φ).

For any ψ ∈ L and any f ∈ A, we shall denote by fψ the element of L defined by :

∀x ∈ H, fψ(x)(z) = f(z)ψ(x)(z).

In particular, (zψ) is defined by (zψ)(x)(z) = z.ψ(x)(z).

Theorem IV.4.1. The map :

\begin{matrix}zR̃ : G & ⟶ 𝔤 \\ ψ & ⟼ zR̃(ψ) \end{matrix}

restricts to a bijection from G^{Φ} onto 𝔤 ∩ L(H, A_{+}).

Proof. For any β ∈ 𝔤 introduce the linear transformation U_{β} of 𝔤 defined by :

U_{β}(A) = β ∗ A + zA ∘ Y.

If β belongs to 𝔤 ∩ L(H, A_{+}) then U_{β} restricts to a linear transformation of 𝔤 ∩ L(H, A_{+}).

Lemma IV.4.2. For any ψ ∈ G, n ∈ ℕ we have :

z^{n}ψ ∘ Y^{n} = ψ ∗ U_{z_{R}(ψ)}^{n}(e).

Proof. Case n = 0 is obvious, n = 1 is just the definition of R̃. We check thus by induction :

z^{n+1}ψ ∘ Y^{n+1} = z(ψ ∘ Y^{n}) ∘ Y

= z(ψ ∗ U_{z_{R}(ψ)}^{n}(e)) ∘ Y

= z(ψ ∘ Y) ∗ U_{z_{R}(ψ)}^{n}(e) + zψ ∗ (U_{z_{R}(ψ)}^{n}(e) ∘ Y )

= ψ ∗ (zR̃(ψ) ∗ U_{z_{R}(ψ)}^{n}(e) + zU_{z_{R}(ψ)}^{n}(e) ∘ Y )

= ψ ∗ U_{z_{R}(ψ)}^{n+1}(e).

•

Let us finish the proof of Theorem IV.4.1 : according to Lemma IV.4.2 we have for any real t, at least formally :

ψ^{t} = ψ ∗ exp(tU_{z_{R}(ψ)})(e).  (∗)

We have still to fix the convergence of the exponential just above in the case when zR̃(ψ) belongs to L(H, A_{+}). Let us consider the following decreasing bifiltration of L(H, A_{+}) :

L_{+}^{p,q} = (z^{q}L(H, A_{+})) ∩ L^{p}.

Considering the associated filtration :

L_{+}^{n} = ⋃_{p+q=n} L_{+}^{p,q},

we see that for any β ∈ 𝔤 ∩ L(H, A_{+}) transformation U_{β} increases filtration by 1, i.e :

U_{β}(L_{+}^{n}) ⊂ L_{+}^{n+1}.

The algebra L(H, A_{+}) is not complete with respect to the topology induced by this filtration, but the completion is L(H, Â_{+}), where Â_{+} = ℂ[[z]] stands for the formal series. Hence the right-hand side of (*) is convergent in L(H, Â_{+}) with respect to this topology. Hence for any γ ∈ L(H, A_{+}) and for ψ such that zR̃(ψ) = γ we have ψ^{t} = ψ ∗ h_{t} with h_{t} ∈ L(H, Â_{+}) for any t. On another hand we already know that h_{t} takes values in meromorphic functions for each t. So h_{t} belongs to L(H, A_{+}).

Then for any β ∈ L(H, A_{+}) and for ψ such that zR̃(ψ) = β we have ψ^{t} = ψ ∗h(t) with h(t) ∈ L(H, A_{+}) for any t. This is equivalent to the fact that ψ belongs to G^{Φ}. Conversely take any ψ in G^{Φ}. Then there exists h_{t} ∈ L(H, A_{+}) such that ψ^{t} = ψ ∗ h_{t}. Hence :

\frac{d}{dt}|_{t=0}ψ_{t} = z(ψ ∘ Y) = ψḣ_{t}|_{t=0}.

So zR̃(ψ) = ḣ_{t}|_{t=0} belongs to L(H, A_{+}). This proves theorem IV.4.1.

•

Lemma IV.4.3.

1) Let ψ ∈ G^{Φ}, and let :

ψ = (ψ_{−})^{∗−1} ∗ ψ_{+}

its Birkhoff decomposition. Then ψ̃ = (ψ_{−})^{∗−1} enjoys property (Φ) as well.

2) Let ψ ∈ G^{Φ} and let h be any element of G without polar part. Then ψ ∗ h enjoys property (Φ) as well.

Proof. Assertion 1) is a consequence of the equality :

ψ^{t} = (ψ^{t})_{−}^{∗−1} ∗ (ψ^{t})_{+} = ψ̃^{t} ∗ (ψ_{+})^{t},

which implies immediately :

ψ̃^{t} = (ψ^{t})_{−}^{∗−1} ∗ (ψ^{t})_{+} ∗ ((ψ_{+})^{t})^{∗−1}.

This is the Birkhoff decomposition of ψ̃^{t} : its polar part is (ψ^{t})_{−}, which by hypothesis is independant of t. Assertion 2) comes from the equality :

(ψ ∗ h)^{t} = ψ^{t} ∗ h^{t} = (ψ^{t})_{−}^{∗−1} ∗ ((ψ^{t})_{+} ∗ h^{t}),

and from the fact that h_{t} does not have any polar part. Thus the right-hand side is the Birkhoff decomposition of (ψ ∗ h)^{t}, the polar part of which is clearly independant of t.

•

Thanks to Lemma IV.4.3 we can now focus our attention on elements ψ of G which enjoy property (Φ) and such that ψ sends Ker ε into A_{−}. Let us denote by G_{−}^{Φ} the subset of G formed by these elements.

Theorem IV.4.4. Let 𝔤^{c} = {β ∈ L(H, ℂ), β(1) = 0}. Let 𝔤_{1}^{c} = 𝔤^{c} ∩𝔤_{1}, and 𝔤_{2}^{c} = 𝔤^{c} ∩𝔤_{2}. Let G_{1,−}^{Φ} = G_{−}^{Φ} ∩G_{1} and G_{2,−}^{Φ} = G_{−}^{Φ} ∩ G_{2}. Then the map :

\begin{matrix}G & ⟶ 𝔤 \\ ψ & ⟼ zR̃(ψ) \end{matrix}

restricts to a bijection from G_{−}^{Φ} (resp. G_{1,−}^{Φ}, G_{2,−}^{Φ}) to 𝔤^{c} (resp 𝔤_{1}^{c}, 𝔤_{2}^{c}) , and we have explicitly for any ψ ∈ G_{−}^{Φ} :

R̃(ψ) = \frac{1}{z}((Res ψ) ∘ Y ).

Proof. We shall need the following key lemma :

Lemma IV.4.5. For any ψ ∈ G_{−}^{Φ} we have :

\frac{d}{dt}|_{t=0}((ψ^{t})_{+}) = Res(ψ ∘ Y).

Proof. Using property (Φ), the fact that ψ(Ker ε) ⊂ A_{−} and the explicit expression of (ψ^{t})_{+} given by theorem II.4.1 we have for any x ∈ H :

(ψ^{t})_{+}(x) = (I − π)(ψ^{t}(x) + ∑_{(x)} ψ^{∗−1}(x′)ψ^{t}(x′′))

= t(I − π)(z|x|ψ(x) + z ∑_{(x)} ψ^{∗−1}(x′)ψ(x′′)|x′′|) + O(t^{2})

= t Res(ψ ∘ Y) + O(t^{2}).

•

Now for any ψ ∈ G_{−}^{Φ} the Birkhoff decomposition of ψ^{t} reads :

ψ^{t} = ψ ∗ (ψ^{t})_{+}.

Differentiating with respect to t at t = 0 we get according to Lemma IV.4.5 :

zψ ∘ Y = ψ ∗ Res(ψ ∘ Y).

We deduce then :

ψ ∘ Y = ψ ∗ \frac{1}{z} Res(ψ ∘ Y),

which proves the equality R̃(ψ) = \frac{1}{z} Res(ψ ∘ Y). As a consequence correspondence zR̃ sends G_{−}^{Φ} into 𝔤^{c}. Conversely let β in 𝔤^{c}. Consider ψ = R̃^{−1}(z^{−1}β). This element of G verifies by definition :

zψ ∘ Y = ψ ∗ β.

Hence for any x ∈ Ker ε we have :

zψ(x) = \frac{1}{|x|}(β(x) + ∑_{(x)} ψ(x′)β(x′′)).

As β(x) is a constant (as a function of the complex variable z) it is easily seen by induction on |x| that the right-hand side evaluated at z has a limit when z tends to infinity. Thus ψ(x) ∈ A_{−}, and then :

ψ = R̃^{−1}(\frac{1}{z}β) ∈ G_{−}^{Φ}.

The proof of the variants of theorem IV.4.4 with G_{1} and G_{2} is then immediate thanks to corollary IV.1.3 and theorem IV.2.1.

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Remark : the “G_{1}” version of Theorem IV.4.4 recovers the result of A. Connes and D. Kreimer in [CK2 § 2] : when ψ is a character, then the ℂ-valued derivation zR̃(ψ) is denoted by β in loc. cit. Theorem IV.4.4 appears then as an interpretation of the result of Connes and Kreimer in other groups than the group of A-valued characters.

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[BF] Ch. Brouder, A. Frabetti, Noncommutative renormalization for massless QED, hepth/0011161 (2000).

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