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. Examples 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 ,··· , 1− 2
∗, {e2πik; k ∈ Q} ∪ {0}, C) := W ae2πi 10s1−1···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. w=w1·w2 swap : (M •, A × B, K) → (M •, B × A, K)
nepar(M •)(w1,.,wr) = (−1)rM (−w1,.,−wr)
Flexions A-semi-group, B-abelian group w = .w1 · w2. w1 = u1 · · · ur , w2 = ur+1 · · · us Symmetries Definition 2. A mould/bimould A• is symmetral (resp. alternal) if ∀w1, w2, Aw = Aw1Aw2 (resp. = 0), w∈sha(w1,w2) 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), w∈she(w1,w2) 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 she(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. Regularization
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 vs1−1 · · · vsr−1.W a(e2πi 10s1−1,··· ,e2πi r 0sr−1)us1−1(u
1 + u2)s2−1 · · · (u1 + u2 + · · · + ur)sr−1.Zag• is symmetral, whereas Zig• satifies another symmetry, symmetril, closely related to symmetrel. Conversion
(mono•, Q/Z × N∗, C) := 1 +
(−1)k−1ζ(k) ki] × Q/Z, C) := miniProposition 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 ARI/GARI
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 w=w2w3w4 M w1 w3N w2 − Nw1 w3M w2 w=w1w2w3 (** 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, w=a1b1c1···bscsas+1 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. IMPORTANT FACT
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 areirreducible 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 ZetaIrrs1,.,sr loma• · · · loma• .

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