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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) 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
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 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• .

Source: http://www.mathematik.uni-muenchen.de/~carr/Kyoto-Moules.pdf