Formal MZVs and

We will now define one of the three main objects of this thesis : the double shuffle Lie algebra𝔡𝔰, originally called𝔡𝔪𝔯₀par Racinet ([R]) for Doubles Mélanges et Régularisation.

Definition I.1.31. Letdenote theℚ-algebra generated by formal symbols𝑍(𝑤)for all words𝑤 subject to the extended double shuffle relations.We call it the algebra offormal MZVs.

For any admissible word𝑤, we again write𝑤=𝑧_𝑠

1…𝑧_𝑠

𝑙 and𝑍(𝑠₁,…, 𝑠_𝑙) =𝑍(𝑤).

Conjecture I.1.32. There is an algebra isomorphism between the algebra of formal MZVs and the algebra generated by real MZVs :

≃.

Definition I.1.33. Following Furusho, we define the space of "new formal zetas"

𝔫𝔣𝔷∶= ()_>2∕(>0)²,

where(>0)²is the ideal generated by products of elements in()of weight at least1. This space is composed of algebraic generators of, called "new" by Furusho since they do not arise as the product of two other MZVs.

We are indeed interested in the structure of𝔫𝔣𝔷as a filtered, graded vector space. We can consider it as an algebra, but its multiplication law is of course trivial.

We will in fact study the graded dual of𝔫𝔣𝔷: called𝔡𝔰, it is obtained by dualizing the defining equations of𝔫𝔣𝔷, which are simply linearized versions of the double shuffle equations. It can be endowed with a Lie algebra structure: the original proof is very complex and due to Racinet ([R]), who named it 𝔡𝔪𝔯₀for Double mélange et régularisation. Its structure can be given more economically in terms of Lie algebra generators than𝔫𝔣𝔷and the dimension of the graded pieces are indeed the same.

Definition I.1.34. Thedouble shuffleLie algebra𝔡𝔰is composed of elements𝑓 ∈ 𝔥of degree at least 3 such that:

𝑐_𝑢⧢𝑣(𝑓) = 0 ∀𝑢, 𝑣∈𝔥 and

𝑐_𝑢∗𝑣(𝑓) = 0 ∀𝑢, 𝑣∈𝔥₁s.𝑡.(𝑢, 𝑣)≠(𝑦^𝑚, 𝑦^𝑛)

where𝑐_𝑤(𝑓)denotes the coefficient of the word𝑤in the polynomial𝑓 (sometimes written(𝑓|𝑤)).

We will see later that the first condition (shuffle) on𝑓 ∈𝔥is equivalent to𝑓 being a Lie polyno-mial. Therefore the definition of𝔡𝔰is sometimes given as:

𝔡𝔰∶= {𝑓 ∈ (𝔩𝔦𝔢(𝑥, 𝑦))_≥3|𝑐_𝑢∗𝑣(𝑓) = 0}

for all𝑢, 𝑣∈𝔥₁such that𝑢, 𝑣are not both powers of𝑦.

Remark I.1.35. Several equivalent definitions of 𝔡𝔰or𝔡𝔪𝔯₀ can be found in the literature. Some of them translate more easily in mould language than others. The original definition by Racinet ([R]) necessitates more background that we are willing to give, but here is the closest form of it used by Schneps:

Definition I.1.36. The Lie algebra 𝔡𝔰is the dual of the Lie coalgebra 𝔫𝔣𝔷of new formal multizeta values. It can be defined directly as the set of polynomials𝑓 ∈𝔥having the two following properties:

(1) The coefficients of f satisfy the shuffle relations :

∑

𝑤∈sℎ(𝑢,𝑣)

(𝑓|𝑤) = 0

where𝑢, 𝑣are words in𝑥, 𝑦and𝑠ℎ(𝑢, 𝑣)is the set of words obtained by shuffling them. This con-dition is equivalent to the assertion that𝑓 ∈𝔩𝔦𝔢₂.

(2) Let𝑓_∗ =𝜋_𝑦(𝑓) +𝑓_{c𝑜𝑟𝑟}, where𝜋_𝑦(𝑓)denotes the projection of𝑓 onto words ending in𝑦and 𝑓_{c𝑜𝑟𝑟} =∑

𝑛≥1

(−1)^𝑛−1

𝑛 (𝑓|𝑥^𝑛−1𝑦)𝑦^𝑛.

(When f is homogeneous of degree n, which we usually assume, then 𝑓_{𝑐𝑜𝑟𝑟} is just the monomial

(−1)^𝑛

𝑛 (𝑓|𝑥^𝑛−1𝑦)𝑦^𝑛.) The coefficients of𝑓_∗satisfy the stuffle relations:

∑

𝑤∈s𝑡(𝑢,𝑣)

(𝑓_∗|𝑤) = 0

where now𝑢, 𝑣and𝑤∈𝔥₁, considered as rewritten in the variables𝑦_𝑖=𝑥^𝑖−1𝑦, and𝑠𝑡(𝑢, 𝑣)is the set of words obtained by stuffling them.

Example I.1.37. The first known non-trivial element of𝔡𝔰is

𝑓₃ = [𝑥,[𝑥, 𝑦]] + [[𝑥, 𝑦], 𝑦] =𝑥²𝑦− 2𝑥𝑦𝑥+𝑦𝑥²+𝑦²𝑥− 2𝑦𝑥𝑦+𝑥𝑦².

Let us first have a look at the shuffle condition : we have to consider all pairs of words𝑢, 𝑣such that some element of their shuffle has non zero coefficient in𝑓.

Let𝑢=𝑥, 𝑣=𝑥𝑦. Then𝑠ℎ(𝑢, 𝑣) = [𝑥²𝑦, 𝑥²𝑦, 𝑥𝑦𝑥]and(𝑓|𝑥²𝑦) + (𝑓|𝑥²𝑦) + (𝑓|𝑥𝑦𝑥) = 1 + 1 − 2 = 0. Similarly, let𝑢 = 𝑥², 𝑣 = 𝑦. Then𝑠ℎ(𝑢, 𝑣) = [𝑥²𝑦, 𝑥𝑦𝑥, 𝑦𝑥²]and(𝑓|𝑥²𝑦) + (𝑓|𝑦𝑥²) + (𝑓|𝑥𝑦𝑥) = 1 + 1 − 2 = 0. The remaining possibilities give the same result and𝑓 verifies the first condition.

To check the stuffle condition, let us first give the explicit expression of𝑓^∗in this case:

𝑓_∗=𝜋_𝑦(𝑓) +𝑓_{𝑐𝑜𝑟𝑟}=𝑥²𝑦− 2𝑦𝑥𝑦+𝑥𝑦²+−𝑦³ 3 . Rewriting it in the variables𝑦_𝑖, it yields

𝑓^∗ =𝑦₃− 2𝑦₁𝑦₂+𝑦₂𝑦₁+ 1 3𝑦³₁. Now here are the two possibles couples𝑢, 𝑣and their associated stuffle :

𝑠𝑡(𝑦₁, 𝑦₂) = [𝑦₁𝑦₂, 𝑦₂𝑦₁, 𝑦₃] gives

(𝑓^∗|𝑦₁𝑦₂) + (𝑓^∗|𝑦₂𝑦₁) + (𝑓^∗|𝑦₃) = −2 + 1 + 1 = 0, and

𝑠𝑡(𝑦₁, 𝑦²₁) = [3𝑦³₁, 𝑦₁𝑦₂, 𝑦₂𝑦₁] yields

(𝑓^∗|3𝑦³₁) + (𝑓^∗|𝑦₁𝑦₂) + (𝑓^∗|𝑦₂𝑦₁) = 1

3⋅3 − 2 + 1 = 0.

Definition I.1.38. We define the Poisson or Ihara Lie bracket on the underlying vector space of𝔩𝔦𝔢₂ by :

{𝑓 , 𝑔} = [𝑓 , 𝑔] +𝐷_𝑓(𝑔) −𝐷_𝑔(𝑓)

where to each element𝑓 ∈ 𝔩𝔦𝔢₂one associates the derivation𝐷_𝑓 such that𝐷_𝑓(𝑥) = 0and𝐷_𝑓(𝑦) = [𝑦, 𝑓].

This bracket corresponds to the Lie bracket on the derivations of 𝔩𝔦𝔢₂ in the sense that 𝐷_{{𝑓 ,𝑔}} = [𝐷_𝑓, 𝐷_𝑔].

Theorem I.1.39. (Racinet, [R]) The double shuffle space𝔡𝔰is a Lie algebra under the Poisson bracket.

The proof contained in Racinet’s thesis is extremely complicated. Salerno and Schneps ([SS]) gave in 2015 an elegant mould theoretic version of this theorem, which we will mention in Chapter 2 .

Note that𝔡𝔰inherits from𝔩𝔦𝔢₂a grading byweightor degree𝑛given by the number of letters𝑥and 𝑦in the Lie word, and a filtration bydepth, the minimal number of𝑦s in a Lie polynomial𝑓. Conjecture I.1.40. The Lie algebra𝔡𝔰is freely generated by one generator of weight𝑛for each odd 𝑛≥3.

The existence of the depth filtration on𝔡𝔰brings the necessity to study the associated graded al-gebra𝑔𝑟(𝔡𝔰), and to see what becomes of the defining property of𝔡𝔰when truncated to their lowest depth part.

The shuffle is an operation that respects depth, therefore the first property remains untouched. How-ever, the stuffle produces terms of the same depth (given by the shuffle) and additional terms of higher depths which disappear when truncating to the lowest depth part (see II.2.8 for an example). This yields the definition of the linearized double shuffle space𝔩𝔰.

Definition I.1.41. The linearized double shuffle space𝔩𝔰is defined to be the set of polynomials𝑓 in 𝑥, 𝑦of degree≥ 3satisfying the shuffle relations (as mentioned earlier, this is equivalent to𝑓 being

in𝔩𝔦𝔢₂) and : ∑

𝑤∈sℎ(𝑢,𝑣)

(𝜋_𝑦(𝑓)|𝑤) = 0

where as above, 𝜋_𝑦(𝑓) is the projection of𝑓 onto the words ending in𝑦, rewritten in the variables 𝑦_𝑖=𝑥_𝑖−1𝑦,𝑢, 𝑣are words in the𝑦_𝑖and𝑤belongs to their shuffle in the alphabet𝑦_𝑖.

However, we exclude from𝔩𝔰all (linear combinations of) the depth 1 even degree polynomials, namely 𝑎𝑑(𝑥)^2𝑛+1(𝑦),𝑛≥1.

The space𝔩𝔰is bigraded by weight and depth, since the shuffle relations respect the depth. We will give a simple mould theoretic proof by Salerno and Schneps of the following result in Chapter 2 : Proposition I.1.42. The space𝔩𝔰^𝑑_𝑛of weight𝑛and depth𝑑is zero if𝑛≠𝑑mod2.

In particular, the graded quotient𝔡𝔰^𝑑_𝑛∕𝔡𝔰^𝑑+1_𝑛 which lies inside it is zero if𝑛≠𝑑mod2.

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