## Mathematik.uni-muenchen.de

**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 **= (

*w*1

*, ., wr*)

*→ M ***w**
*N • *: (

*A × B*)

*∗ → K*
1

*, · · · , r → N***w***.*
**Examples**
*Ze•∗ *is the bimould defined by

*∗, *Q

*/*Z

*× *N

*∗, *C) :=

*Ze∗*
*n*1

*>n*2

*>···nr>*0
with

*s*1

*≥ *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

*∗, {e*2

*πik*;

*k ∈ *Q

*} ∪ {*0

*}, *C) :=

*W ae*2

*πi *10

*s*1

*−*1

*···e*2

*πi r *0

*sr−*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• *:

*C***w **=

*M ***w **+

*N ***w**
*M • × N• *=

*mu*(

*M •, N •*) =

*C• *:

*C***w **=

*M ***w**1

*· N***w**2

*.*
**w**=

**w**1

*·***w**2

*swap *: (

*M •, A × B, K*)

*→ *(

*M •, B × A, K*)

*nepar*(

*M •*)(

*w*1

*,.,wr*) = (

*−*1)

*rM *(

*−w*1

*,.,−wr*)

*Flexions*

A-semi-group,

*B*-abelian group

**w **=

*.***w**1

*· ***w**2

*.*

**w**1 =

*u*1

*· · · ur , ***w**2 =

*ur*+1

*· · · us*
**Symmetries**
**Definition 2. ***A mould/bimould A• is symmetral (resp. alternal) if*
*∀***w**1

*, ***w**2

*,*
*A***w **=

*A***w**1

*A***w**2

*(resp. *= 0)

*,*
**w***∈sha*(

**w**1

*,***w**2)

*where sha*(

**w**1

*, ***w**2)

*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*
*∀***w**1

*, ***w**2

*,*
*A***w **=

*A***w**1

*A***w**2

*(resp. *= 0)

*,*
**w***∈she*(

**w**1

*,***w**2)

*where she*(

**w**1

*, ***w**2)

*denotes the “contracting shuffle” or “stuffle” product of sequences, which is given by the*

recursion relation,
**w**1 = (

*a*1

*, ., ar*)

*, ***w**2 = (

*ar*+1

*, ., ar*+

*s*)

*she*(

**w***i, ∅*) =

**w***i*

she(

**w**1

*, ***w**2) =

*a*1

*· she*((

*a*2

*, ., ar*)

*, ***w**2)

*ar*+1

*· she*(

**w**1

*, *(

*ar*+2

*, ., ar*+

*s*))

(

*a*1 +

*ar*+1)

*· she*((

*a*2

*, ., 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

*{e*2

*πik*;

*k ∈ *Q

*} ∪ {*0

*}*,

*· W a*(1) =

*W a*(0) = 0,

*· W a• *is symmetral.

**Generating Series**
*vs*1

*−*1

*· · · vsr−*1

*.*
*W a*(

*e*2

*πi *10

*s*1

*−*1

*,··· ,e*2

*πi r *0

*sr−*1)

*us*1

*−*1(

*u*
1 +

*u*2)

*s*2

*−*1

*· · · *(

*u*1 +

*u*2 +

*· · · *+

*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−*1

*ζ*(

*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 **=

*A***w**1

*B***w**2

*− B***w**1

*A***w**2 +

*M ***w**3

*N ***w**2

**w**4

*− N ***w**3

*M ***w**2

**w**4

**w**=

**w**2

**w**3

**w**4

*M ***w**1

**w**3

*N ***w**2

*− N***w**1

**w**3

*M ***w**2

**w**=

**w**1

**w**2

**w**3

*(** 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 ***B***a1 ***· · · B***as**+

**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, *Z

*eta;*
*• The irreducibles appearing as coefficients in the factors give us a factorization for the multiple zeta value*
Z

*eta *:= Z

*etaI ⊗ *Z

*etaII ⊗ *Z

*etaIII.*
*· *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, Z

*etaIII*, is generated by irreducibles of odd depth,
i.e. linear combinations of

*ζ*(

*s*1

*, ., sr*) where

*r *is odd. The mould of such irreducibles is denoted by

*Irr• *.

*· *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

*Irrs*1

*,.,sr loma• · · · loma• ,*
where

*loma• *is the restriction of

*loma• *to the weight

*s*
*· Zag***w ***∈ *Q[[

*u*
1 = 0), which in the language of zetas, means that the corresponding factor in
the Z

*eta *algebra, Z

*etaI*, is generated by 6

*ζ*(2) =

*π*2.

*· *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 Z

*eta*
*Irrs*1

*,.,sr loma• · · · loma• .*
Source: http://www.mathematik.uni-muenchen.de/~carr/Kyoto-Moules.pdf

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