Definition 1. The standard definition is that a mould is a function on “a variable number of variables”. To
flesh out this definition, in the general case, let A, B be sets and K be an algebra. A
mould, M • = (M •, A, K),
is a map from the free monoid A∗ into K and a bimould is defined as a function on the free monoid of the
Cartesian product of two sets,
(A × B)∗:
w = (w1, ., wr) → M w
N • : (A × B)∗ → K 1 , · · · , r → Nw.
Ze•∗ is the bimould defined by ∗, Q/Z × N∗, C) := Ze∗ n1>n2>···nr>0 with s1 2. If we take i = 0 ∀i then we obtain the usual multiple zeta values. Sometimes people say thatelements in the image of this mould are “colored multiple zeta values”.
r , r−1− r ,··· , 12 ∗, {e2πik; k ∈ Q} ∪ {0}, C) := W ae2πi 10s11···e2πi r 0sr−1 Hence we require that the first term be a root of unity and the last term be 0.
Operations on Moulds
Given two moulds (resp. bimoulds) (M •, A(resp. × B), K) and (N •, A(resp. × B), K) addition and multi-plication are given by M • + N • = C• : Cw = M w + N w
M • × N• = mu(M •, N •) = C• : Cw =
M w1 · Nw2.
swap : (M •, A × B, K) (M •, B × A, K) nepar(M •)(w1,.,wr) = (1)rM (−w1,.,−wr) Flexions
-semi-group, B-abelian group
w = .w1 · w2.
w1 =
u1 · · · ur , w2 = ur+1 · · · us
Definition 2. A mould/bimould A• is symmetral (resp. alternal) if
w1, w2,
Aw = Aw1Aw2 (resp. = 0),
where sha(w1, w2) denotes the shuffle product of sequences. We say such a mould is “as” (resp. “al”). W a•∗
is as.

A mould/bimould A• is symmetrel (resp. alternel) if w1, w2,
Aw = Aw1Aw2 (resp. = 0),
where she(w1, w2) denotes the “contracting shuffle” or “stuffle” product of sequences, which is given by the
recursion relation,

w1 = (a1, ., ar), w2 = (ar+1, ., ar+s)
she(wi, ∅) = wi
(w1, w2) = a1 · she((a2, ., ar), w2)
ar+1 · she(w1, (ar+2, ., ar+s))
(a1 + ar+1) · she((a2, ., ar), (ar+2, ., ar+s)). Such a mould is called es (resp. el). Ze•∗ is es.
More Examples
The following examples of moulds define two generating series for multiple zeta values, and provide a methodof regularization of multiple zeta values.
There exists a unique mould, Ze•, such that · Ze• = Ze•∗ wherever Ze•∗ is defined,· Ze• is defined on all of (Q/Z × N) Likewise, there exists a unique mould, W a•, such that · W a• = W a•∗ wherever Wa•∗ is defined,· W a• is defined on all {e2πik; k ∈ Q} ∪ {0},· W a(1) = W a(0) = 0,· W a• is symmetral.
Generating Series
vs11 · · · vsr−1. W a(e2πi 10s11,··· ,e2πi r 0sr−1)us11(u 1 + u2)s21 · · · (u1 + u2 + · · · + ur)sr−1. Zag• is symmetral, whereas Zig• satifies another symmetry, symmetril, closely related to symmetrel.
(mono•, Q/Z × N∗, C) := 1 + (1)k−1ζ(k) k i] × Q/Z, C) := mini Proposition 3. mini• × swap(Zag•) = Zig•
We say then that Zag• is as/is, since it’s symmetral and its swap is symmetril (up to multiplication by Key Results
In order to keep simplicity for this talk, we take the following definition, which is more restricted than theusual general definition.
Definition 4. Let ARIal/il be the Lie algebra with the following definition.
• As a vector space over Q, (ARIal/il, Q[ui] × Q/Z, C[[ui]]) := M • : M ∅ = 0, M is al, swap(M •) is alternil∗∗ , ari(M •, N •)w =
Aw1Bw2 − Bw1Aw2 +
M w3N w2 w4 − N w3M w2 w4
M w1 w3N w2 − Nw1 w3M w2
(** The alternil condition means “up to a multiplication by a commutative bimould”.) The ari bracket is equal to the Lie-Poisson bracket {, }, up to a variable change, on the usual Lie algebra of multizeta values (dm). However, the ari bracket can be defined on a more general set of vector spaces,which is a tool Ecalle uses in his proofs.
The flexions ( , , , ) in the definition of the “ari” bracket correspond to the derivations, Df(x) = 0, Df (y) = [y, f ], which give the definition of the above mentioned Poisson bracket.
Definition 5. By taking the ari-exponential of the Lie algebra, ARIal/il, we obtain a Lie group, GARIas/is
which has the following presentation.

(GARIas/is, Q[ui] × Q/Z, C[[ui]]) := {M • : M ∅ = 1, M is as, swap(M •) is symmetril∗∗}, gari(A•, B•)w =
A b1 · · · A bs Ba1 · · · Bas+1 (B−1) c1 · · · (B−1) cs,
where s 0, bi = ∅ (∀1 i s), ci · ai+1 = ∅ (∀1 i s 1) and (B−1) denotes the inverse for
standard mould multiplication.

• invgari(M•) is inverse of a mould M• for the gari product, gari(invgari(A•), A•) = gari(A•, invgari(A)) = 1•, where 1= 1, 1w = 0.
The mould Zag• is an element of the Lie group, GARI.
Canonical Decomposition into Irreductibles
Theorem 6. The mould Zag• may be decomposed into three factors,
Zag• = gari(Zag•, Zag• , Zag• ) • The even/odd length components of Zag• are even/odd functions of w, while the even/odd legth
are odd/even functions of w;
• Each component is decomposed as a series in a basis of ARIal/il, which when evaluated at i = 0 are irreducible elements of the Q algebra of multiple zeta values, Zeta; • The irreducibles appearing as coefficients in the factors give us a factorization for the multiple zeta value Zeta := ZetaI ⊗ ZetaII ⊗ ZetaIII. · The factor Zag• is the most simple to express explicitly, gari(Zag• , Zag• ) = gari(nepar(invgari(Zag•)), Zag•). By linearizing, you can see that indeed this provides an odd/even function on components of even/oddlength.
· The length 1 component is given by · The associated factor in the multiple zeta value algebra, ZetaIII, is generated by irreducibles of odd depth, i.e. linear combinations of ζ(s1, ., sr) where r is odd. The mould of such irreducibles is denoted byIrr• .
· We get an explicit expression for the set of irreducible multiple zeta values in factor Zag• in terms of a mould loma•, which is a generating mould which (vaguely speaking) forms a basis of rational polynomialsfor ARIal/il (the explicit construction is out of the scope of this talk). We have Irrs1,.,sr loma• · · · loma• , where loma• is the restriction of loma• to the weight s · Zagw Q[[u
1 = 0), which in the language of zetas, means that the corresponding factor in the Zeta algebra, ZetaI, is generated by 6ζ(2) = π2.
· The explicit factorization of Zag• from Zag• is a very costly analytic contruction, whose difficulty comes from getting rid of unwanted singularities. The formula is the following: Zag• = gari(tal•, invgari(pal•), expari(roma•)), and again the definition of pal and roma go out of the scope, since they are very long.
· Zag• is explicitly calculated by factoring Zag• by Zag• and Zag• .
· Zag• may be factored as a generating series for the irreducible multiple zeta values of even depth in the same manner as Zag• , providing a set of “canonical” irreducibles for Zeta Irrs1,.,sr loma• · · · loma• .


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