The dihedral Lie algebras and Galois symmetries of π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞})

A.B. Goncharov

1 Introduction

1. Summary. In this paper we study the action of the absolute Galois group on the pro-l completion π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}) of the fundamental group of ℙ^{1} minus 0, ∞ and N-th roots of unity using the following tools.

Let G be a commutative group. In [G1] we constructed a bigraded Lie algebra 𝔻_{•,•}(G), called the dihedral Lie algebra of G, see section 2 below. Let μ_{N} be the group of N-th roots of unity. We relate 𝔻_{•,•}(μ_{N}) to the Lie algebra of the image of Gal(\overline{ℚ}/ℚ) in Autπ_{1}^{(l)}(ℙ^{1}\\{0, μ_{N}, ∞\}). On the other hand, in [G1] we found a mysterious connection between the structure of the Lie algebra 𝔻_{•,•}(μ_{N}) and geometry of the quotient of the symmetric space GL_{m}(ℝ)/R^{∗} · SO_{m} by the subgroup Γ_{1}(m; N) ⊂ GL_{m}(ℤ) stabilizing the vector (1, 0, ..., 0) mod N. Using these results we investigate the action of Gal(\overline{ℚ}/ℚ) on π_{1}^{(l)}(ℙ^{1}\\{0, μ_{N}, ∞\}). In particular we obtain new results about the action on π_{1}^{(l)}(ℙ^{1}\{0, 1, ∞}).

2. The action of the Galois group. Let X be a regular curve over a number field, \overline{X} the corresponding projective curve, and v a tangent vector at a point x ∈ \overline{X}. Then according to Deligne ([D]) one can define the profinite fundamental group π̂_{1}(X, v) based at the vector v. If X, x and v are defined over a field L then the group Gal(\overline{ℚ}/L) acts by automorphisms of π_{1}^{(l)}(X, v).

If X = ℙ^{1}\{0, μ_{N}, ∞} there is a tangent vector v_{∞1} corresponding to the inverse t^{−1} of the canonical coordinate t on ℙ^{1}\{0, μ_{N}, ∞}. So we get a map

Φ_{N}^{(l)} : Gal(\overline{ℚ}/ℚ) ⟶ Autπ_{1}^{(l)}(ℙ^{1}\\{0, μ_{N}, ∞\}, v_{∞1})  (1)

There are canonical isomorphisms ℤ_{l}(1) = π_{1}^{(l)}(T_{0}ℙ^{1}, v_{∞1}) = π_{1}^{(l)}(ℙ^{1}\{0, ∞}, v_{∞1}). We have a projection

π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1}) ⟶ π_{1}^{(l)}(ℙ^{1}\{0, ∞}, v_{∞1})  (2)

The tangent vector v_{∞1} provides a splitting of this projection:

I_{∞} : π_{1}^{(l)}(ℙ^{1}\{0, ∞}, v_{∞1}) ↪ π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1})  (3)

It commutes with the action of the Galois group. Let ζ_{n} be a primitive n’s root of unity, so ℚ(ζ_{n}) is the field generated by the n’s roots of unity. Then it follows that the subgroup Gal(\overline{ℚ}/ℚ(ζ_{l^{∞}})) preserves the elements of the subgroup

I_{∞}(ℤ_{l}(1)) ⊂ π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞})  (4)

The restriction of the map Φ_{N}^{(l)} to the subgroup Gal(\overline{ℚ}/ℚ(ζ_{N}, ζ_{l^{∞}})) satisfies some additional constraints. Indeed, the subgroup Gal(\overline{ℚ}/ℚ(ζ_{N})) preserves each of the points 0, ζ_{N}^{a}, ∞ on ℙ^{1} and commutes with the natural action of the group μ_{N} on ℙ^{1}\{0, μ_{N}, ∞} given by z ⟼ ζ_{N}z.

So we conclude that Gal(\overline{ℚ}/ℚ(ζ_{N}, ζ_{l^{∞}})) preserves the elements of I_{∞}(ℤ_{l}(1)), the conjugacy classes of the loops around 0 and N-th roots of unity, and commutes with the action of the group μ_{N}.

3. Passing to the Lie algebras. Let 𝕃(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1}) be the pronilpotent Lie algebra over ℚ corresponding by the Maltsev theory to the pronilpotent completion of the fundamental group π_{1}(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1}) (see [D] ch. 9). We use a shorthand 𝕃_{N} for it. In particular we have the following.

i) Let 𝕃_{N}^{(l)} be the Lie algebra over ℚ_{l} corresponding to the pro-l group π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1}). Then 𝕃_{N}^{(l)} = 𝕃_{N} ⊗_{ℚ} ℚ_{l}. Let Exp(𝕃_{N}) be the connected, simply-connected prounipotent algebraic group over ℚ with the Lie algebra 𝕃_{N}; then π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1}) is a compact open subgroup of Exp(𝕃_{N})(ℚ_{l}).

ii) 𝕃_{N} is isomorphic to the pronilpotent completion of a free Lie algebra with N + 1 generators corresponding to the loops around 0 and all N-th roots of unity.

Similar to (2) and (3) there is a canonical projection

p : 𝕃(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1}) ⟶ 𝕃(ℙ^{1}\{0, ∞}, v_{∞1})  (5)

and the tangent vector v_{∞1} provides its splitting:

i_{∞} : 𝕃(ℙ^{1}\{0, ∞}, v_{∞1}) = 𝕃(T_{0}ℙ^{1}, v_{∞1}) = ℚ(1) ↪ 𝕃(ℙ^{1}\{0, μ_{N}, ∞}, v_{∞1})

Denote by X_{∞} the corresponding generator of 𝕃_{N}. Choose generators X_{0} and X_{ζ} corresponding to “loops” around 0 and ζ ∈ μ_{N} in such a way that X_{0} + ∑_{ζ∈μ_{N}} X_{ζ} + X_{∞} = 0 and the action of μ_{N} on ℙ^{1} permutes the generators X_{ζ}.

There is a homomorphism φ_{N}^{(l)} : Gal(\overline{ℚ}/ℚ) ⟶ Aut𝕃_{N}^{(l)}. Denote by G_{N}^{(l)} the Lie algebra of the image of the subgroup Gal(\overline{ℚ}/ℚ(ζ_{N}, ζ_{l^{∞}})) under this map. It acts by derivations of the Lie algebra 𝕃_{N}^{(l)}. We will describe the constraints on the derivations we get using a more general set up presented below.

4. Special equivariant derivations. Let G be a commutative group written multiplicatively. Let 𝕃(G) be the pronilpotent completion of the free Lie algebra with the generators Y and X_{g} where g runs through the group G.

Example. The Lie algebra 𝕃(μ_{N}) is isomorphic to 𝕃_{N}.

A derivation D of the Lie algebra 𝕃(G) is called special if there are elements S, S_{g} ∈ 𝕃(G) such that

D(X_{g}) = [S_{g}, X_{g}] for any g ∈ G, D(Y) = [S, Y] and D(Y + ∑_{g∈G} X_{g}) = 0

The group G acts on the generators by h : Y ⟼ Y, X_{g} ⟼ X_{hg}. So it acts by automorphisms of the Lie algebra 𝕃(G). A derivation D of 𝕃(G) is called equivariant if it commutes with the action of G. Let Der^{SE}𝕃(G) be the Lie algebra of all special equivariant derivations of the Lie algebra 𝕃(G).

There are two increasing filtrations by ideals on the Lie algebra 𝕃(G), going from −∞.

The weight filtration F_{•}^{W}. It coincides with the lower central series for 𝕃(G):

𝕃(G) = F_{−1}^{W}𝕃(G); F_{−n−1}^{W}𝕃(G) := [F_{−n}^{W}𝕃(G), 𝕃(G)]

The depth filtration F_{•}^{D}. Let I(G) be the the kernel of the projection p_{G} : 𝕃(G) ⟶ ℚ, X_{g} ⟼ 0, Y ⟼ 1. Its powers give the depth filtration:

F_{0}^{D}𝕃(G) = 𝕃(G), F_{−1}^{D}𝕃 = I(G), F_{−n−1}^{D}𝕃(G) = [I(G), F_{−n}^{D}𝕃(G)]

The weight (resp. depth) of a Lie word in Y, X_{g} is (−1)× the total number of the generators (resp. the number of X_{g}’s).

Example. The depth filtration on 𝕃(μ_{N}) is provided by the inclusion ℙ^{1}\{0, μ_{N}, ∞} ⟶ ℙ^{1}\{0, ∞} which induces the projection p_{μ_{N}}.

These filtrations induce two filtrations on the Lie algebra Der^{SE}𝕃(G). Taking the associated graded quotient with respect to these filtrations we get a Lie algebra GrDer_{•,•}^{SE}𝕃(G) bigraded by the weight −w and depth −m.

Now let us put G = μ_{N} and return to the problem of description of the map φ_{N}^{(l)}. It follows from s. 1.2 that G_{N}^{(l)} acts by special equivariant derivations of the Lie algebra 𝕃_{N}^{(l)}, i.e. G_{N}^{(l)} ⊂ Der^{SE}𝕃_{N}^{(l)} = Der^{SE}𝕃_{N} ⊗ ℚ_{l}. The map φ_{N}^{(l)} obviously respects the two filtrations. The filtrations on Der^{SE}𝕃_{N}^{(l)} induce two filtrations on G_{N}^{(l)}. Let

GrG_{•,•}^{(l)}(μ_{N}) ⊂ GrDer_{•,•}^{SE}𝕃(G)  (6)

be the associated graded quotient of the Lie algebra G_{N}^{(l)}.

Remark. The weight filtration can be defined on the Lie algebra attached to the pronilpotent completion of the fundamental group of an arbitrary algebraic variety. The weights on 𝕃(μ_{N}) are obtained by dividing by 2 the usual weights.

5. Main tools and conjectures. The goal of the present paper is to study Lie algebra (6). Our main tool is the following result proved in section 3

Theorem 1.1 Let G be a finite commutative group. Then there exists an injective morphism of bigraded Lie algebra

ξ_{G} : 𝔻_{•,•}(G) ↪ GrDer_{•,•}^{SE}𝕃(G)

Conjecture 1.2 GrG_{•,•}^{(l)}(μ_{N}) ⊂ ξ_{μ_{N}} (𝔻_{•,•}(μ_{N})) ⊗_{ℚ} ℚ_{l}.

This conjecture is essentially proved in this paper, see s. 4.5.

If G is a trivial group we set 𝔻_{•,•} := 𝔻_{•,•}({e}), G_{•,•}^{(l)} := G_{•,•}^{(l)}(1) and ξ := ξ_{{e}}.

Conjecture 1.3 One has ξ(𝔻_{•,•}) ⊗ ℚ_{l} = GrG_{•,•}^{(l)}.

Since the Lie algebra 𝔻_{•,•} is defined very explicitely, this would give a precise description of the image of the Galois group, and so seems to be very hard. The results of Soulé [So], Ihara [Ih2] and Deligne [D] imply it in the depth -1. In this paper we proved that conjecture 1.3 holds in the depth -2 and -3 if we assume conjecture 1.2.

6. Some results about the action of the absolute Galois group on π_{1}^{(l)}(ℙ^{1}\{0, 1, ∞}, v_{∞1}). For every m ≥ 1 we have a nilpotent graded Lie algebra

GrG_{•,≥−m}^{(l)} := Gr (G_{•,•}^{(l)}/G_{•,<−m}^{(l)})

It is of the nilpotence class m. In particular GrG_{•,≥−1}^{(l)} is abelian. It follows from the mentioned above results of Soulé, Ihara and Deligne that

GrG_{−w,−1}^{(l)} = \{ \begin{matrix}0 & w even, or w = 1 \\ ℚ_{l} & w > 1 odd \end{matrix}  (7)

Let us try to go further. Let L_{−•,−k}^{(l)} ⊂ GrG_{−•,−k}^{(l)} be the subspace generated

by commutators of k elements of GrG_{−•,−1}^{(l)}. Then L_{−•,≥−m}^{(l)} := ⊕_{k=1}^{m}L_{−•,−k}^{(l)}

is a Lie subalgebra of GrG_{−•,≥−m}^{(l)}. The structure of the Lie algebra L_{•,≥−2}^{(l)} is

described by the commutator map [,] : Λ^{2}L_{−•,−1}^{(l)} ⟶ L_{−•,−2}^{(l)}. Dualizing it we get the standard cochain complex of this Lie algebra:

L_{−•,−2}^{(l)}^{∨} ⟶^{δ} Λ^{2}L_{−•,−1}^{(l)}^{∨}  (8)

On the other hand, let 𝕄_{(2)}^{∗} := 𝕄_{(2)}^{1} ⟶ 𝕄_{(2)}^{2} ⟶ 𝕄_{(2)}^{3} be the chain complex of the classical modular triangulation of the hyperbolic plane extended by ℙ^{1}(ℚ), placed in degrees [1, 3].

For example 𝕄_{(2)}^{1} is the group generated by the triangles. The group GL_{2}(ℤ) acts naturally on the hyperbolic plane preserving the modular picture. So 𝕄_{(2)}^{∗} is a complex of GL_{2}(ℤ)-modules. Let V be a GL_{2}-module. Then the complex 𝕄_{(2)}^{∗} ⊗_{GL_{2}(ℤ)} V [1] computes the cuspidal cohomology H_{cusp}^{∗}(GL_{2}(ℤ), V). Let

τ_{[1,2]}(𝕄_{(2)}^{∗} ⊗_{GL_{2}(ℤ)} V) :=

𝕄_{(2)}^{1} ⊗_{GL_{2}(ℤ)} V ⟶ Ker(𝕄_{(2)}^{2} ⊗_{GL_{2}(ℤ)} V ⟶ 𝕄_{(2)}^{3} ⊗_{GL_{2}(ℤ)} V )  (9)

Theorem 1.4 a) The weight w part of the complex (8) is canonically isomorphic to the complex τ_{[1,2]}(𝕄_{(2)}^{∗} ⊗_{GL_{2}(ℤ)} S^{w−2}V_{2}) ⊗ ℚ_{l}.

b)  dimGrG_{−w,−2}^{(l)} ≥ dimL_{−w,−2}^{(l)} = \{ \begin{matrix}0 & w : odd \\ [\frac{w−2}{6}] & w : even \end{matrix}  (10)

c) Let us assume conjecture 1.2 for N = 1. Then L_{−w,−2}^{(l)} = GrG_{−w,−2}^{(l)}.

The part b) has been also obtained by Ihara and Takao (unpublished, cf [M]). In particular there is a canonical isomorphism

m_{1}^{(l)} : L_{−w,−2}^{(l)}^{∨} ⟶^{=} 𝕄_{(2)}^{1} ⊗_{GL_{2}(ℤ)} S^{w−2}V_{2} ⊗ ℚ_{l}

Denote by Λ_{w}^{2} the weight w part of Λ^{2}. We define an isomorphism

m_{2}^{(l)} : Λ_{w}^{2}(L_{•,−1}^{(l)})^{∨} ⟶^{=} 𝕄_{(2)}^{2} ⊗_{GL_{2}(ℤ)} S^{w−2}V_{2} ⊗ ℚ_{l}  (11)

as follows. The stabilizer in GL_{2}(ℤ) of the path from 0 to i∞ on the upper

half plane is generated by ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ). Therefore the right hand

side of (11) is identified with the space of degree w − 2 polynomials in t_{1}, t_{2}, skewsymmetric in the variables t_{1}, t_{2} and of even degree in each of them. Let

ζ_{M}^{(l)}(n) be Soule’s generator of (GrG_{−2n+1,−1}^{(l)})^{∨}, see section 5. Then

m_{2}^{(l)} : ζ_{M}^{(l)}(2m + 1) ∧ ζ_{M}^{(l)}(2n + 1) ⟼ t_{1}^{2m}t_{2}^{2n} − t_{1}^{2n}t_{2}^{2m}

We obtain similar results in the depth three case. The depth three part of the standard cochain complex of the Lie agebra L_{−•,≥−3}^{(l)} looks as follows.

L_{−•,−3}^{(l)}^{∨} ⟶ L_{−•,−2}^{(l)} ⊗ L_{−•,−1}^{(l)}^{∨} ⟶ Λ^{3}L_{−•,−1}^{(l)}^{∨}  (12)

The first map is dual to the commutator map [,] : L_{−•,−2}^{(l)} ⊗ L_{−•,−1}^{(l)} ⟶ L_{−•,−3}^{(l)}.

Theorem 1.5 a) The complex (12) computes the cuspidal cohomology H_{cusp}^{i}(GL_{3}(ℤ), S^{w−3}V_{3}) at i = 1, 2, 3. Therefore it is acyclic.

b)  dimGrG_{−w,−3}^{(l)} ≥ dimL_{−w,−3}^{(l)} = \{ \begin{matrix}0 & w : even \\ [\frac{(w−3)^{2}−1}{48}] & w : odd \end{matrix}  (13)

c) Let us assume conjecture 1.2 for N = 1. Then GrG_{−w,−3}^{(l)} = L_{−w,−3}^{(l)}.

To prove this theorem we combine the results of section 4 with the main result of [G1] to show that complex (12) is naturally isomorphic to a certain subcomplex of the Voronoi complex computing cuspidal cohomology at i = 1, 2, 3 for the local system on GL_{3}(ℤ)\GL_{3}(ℝ)/O(3) associated with S^{w−3}V_{3}.

Remarks. a) Compare theorems 1.4, 1.5 with theorems 1.4, 1.5 in [G1] where we have proved that the ℚ-space of reduced multiple ζ-values of depth 2 and 3 is estimated from above by (10) and (13).

b) We hope to show that GrG_{−w,−4}^{(l)}/L_{−w,−4}^{(l)} = H_{cusp}^{1}(GL_{2}(ℤ), S^{w−2}V_{2}) ⊗ ℚ_{l} in the subsiquent paper.

One has GrDer_{−w,−m}^{SE}𝕃(G) ≠ 0 only if w ≥ m ≥ 1, and similar constraints hold for GrG^{(l)}(μ_{N}) and 𝔻_{•,•}(G). We define the diagonal Lie algebras 𝔻_{•}^{diag}(G) and GrG_{•}^{(l)}(μ_{N}) as the quotients of 𝔻_{•,•}(G) and GrG_{•,•}^{(l)}(μ_{N}) by the components 𝔻_{−w,−m}(G) and GrG_{−w,−m}^{(l)}(μ_{N}) with w ≠ m. So they are graded by the weight.

Conjecture 1.6 Let p be a prime number. Then ζ_{μ_{p}}(𝔻_{•}^{diag}(μ_{p})) = GrG_{•}^{(l)}(μ_{p}).

The results of this paper combined with [G3] prove this for the weights ≥ −2. Acknowledgment. I am grateful to the Max-Planck-Institute (Bonn) for

hospitality and support. The support by the NSF grant DMS-9800998 is gratefully acknowledged. I am grateful to M. Kontsevich for very useful discussions.

2 The dihedral Lie coalgebra of a commutative group G

1. Definitions. First let us recall a definition of a bigraded Lie coalgebra D_{•,•}(G) = ⊕_{w≥m≥1}D_{w,m}(G) given in [G1]. The ℚ-vector space D_{w,m}(G) is generated by the symbols

{g_{0} : ... : g_{m}}_{n_{1},...,n_{m}},  w = n_{1} + ... + n_{m}, n_{i} ≥ 1  (14)

To define the relations we introduce the generating functions

{g_{0} : ... : g_{m}|t_{0} : .... : t_{m}} := ∑_{n_{i}>0}{g_{0} : ... : g_{m}}_{n_{1},...,n_{m}}(t_{1}−t_{0})^{n_{1}−1}...(t_{m}−t_{0})^{n_{m}−1}

(15)

Notice that {g_{0} : ... : g_{m}|t_{0} : ... : t_{m}} = {g_{0} : ... : g_{m}|t + t_{0} : ... : t + t_{m}} (16)

We picture the generating functions (15) as m + 1 pairs (g_{0}, t_{0}), ..., (g_{m}, t_{m}), called an extended dihedral word, located cyclically on an oriented circle (see the picture on page 7).

We will need also two other generating functions:

{g_{0} : g_{1} : ... : g_{m}|t_{0}, ..., t_{m}} := {g_{0} : ... : g_{m}|t_{0} : t_{0} + t_{1} : ... : t_{0} + ... + t_{m}} (17)

where t_{0} + ... + t_{m} = 0, and

{g_{0}, g_{1}, ..., g_{m}|t_{0} : ... : t_{m}} := {g_{0} : g_{0}g_{1} : ... : g_{0}...g_{m}|t_{0} : t_{1} : ... : t_{m}}

where g_{0} · ... · g_{m} = 1.

Relations. i) For any g ∈ G one has

{g · g_{0} : ... : g · g_{m}|t_{0} : ... : t_{m}} = {g_{0} : ... : g_{m}|t_{0} : ... : t_{m}}  (18)

ii) The dihedral symmetry relations:

{g_{0} : ... : g_{m−1} : g_{m}|t_{0} : t_{1} : ... : t_{m}} = {g_{1} : ... : g_{m} : g_{0}|t_{1} : ... : t_{m} : t_{0}} (19)
{g_{0} : ... : g_{m}|t_{0} : ... : t_{m}} = (−1)^{m+1}{g_{m} : ... : g_{0}|t_{m} : ... : t_{0}}  (20)

iii) The double shuffle relations (k + l = m, k ≥ 1, l ≥ 1):

∑_{σ∈Σ_{k,l}}{g_{0} : g_{σ(1)} : ... : g_{σ(m)}|t_{0}, t_{σ(1)}, ..., t_{σ(m)}} = 0,  (21)

∑_{σ∈Σ_{k,l}}{g_{0}, g_{σ(1)}, ..., g_{σ(m)}|t_{0} : t_{σ(1)} : ... : t_{σ(m)}} = 0  (22)

here Σ_{k,l} is the set of all shuffles of the ordered sets {1, ..., k} and {k+1, ..., k+l}. iv) The distribution relations (l ∈ ℤ and |l| divides |G| if the group is finite.)

{x_{0}^{l} : ... : x_{m}^{l}|t_{0} : t_{1} : ... : t_{m}} − ∑_{y_{i}^{l}=x_{i}^{l}}{y_{0} : ... : y_{m}|l · t_{0} : ... : l · t_{m}} = 0

except the relation {e : e}_{1} = ∑_{y^{l}=e}{y : e}_{1}, which is not supposed to hold. v) {e : e}_{1} = 0. Remark. In [G1] we did not impose condition v). For example the distribution relations for l = −1 are

{x_{0}^{−1} : ... : x_{m}^{−1}|t_{0} : t_{1} : ... : t_{m}} = {x_{0} : ... : x_{m}| − t_{0} : ... : −t_{m}} (23)

It was proved in [G1] that the dihedral symmetry and (23) follow from the double shuffle relations, see the proof of theorem 4.1 in [G1].

The cobracket δ : D_{•,•}(G) ⟶ D_{•,•}(G) ∧ D_{•,•}(G) is given by

δ{g_{0} : ... : g_{m}|t_{0}, ..., t_{m}} :=  (24)

∑_{k=1}^{m−1} Cycle_{m+1}({g_{0} : ... : g_{k−1} : g_{k}|t_{0}, ..., t_{k−1}, x_{k}}∧{g_{k} : g_{k+1} : ... : g_{m}|y_{k}, t_{k+1}, ..., t_{m}})

where t_{0}+...+t_{k−1}+x_{k} = 0, y_{k}+t_{k+1}+...+t_{m} = 0, the indices are modulo m+1 and Cycle_{m+1}f(x_{0}, ..., x_{m}) := ∑_{i=0}^{m} f(x_{i}, ..., x_{m+i}). Each term of the formula corresponds to the following procedure: choose an arc on the circle between the two neighboring pairs (g_{α}, t_{α}) and (g_{α+1}, t_{α+1}), and in addition choose a pairs (g_{β}, t_{β}) different from the ends of the arc. Then cut the circle in the choosen arc and in the choosen point and make two oriented circles with an extended dihedral word on each of them out of the initial data.

Theorem 2.1 δ provides a bigraded Lie coalgebra structure on D_{•,•}(G).

We will need below an explicit description of the coproduct on the generating series {g_{0} : ... : g_{m}|u_{0} : ... : u_{m}}. Lemma 2.2 One has

δ{g_{0} : ... : g_{m}|u_{0} : ... : u_{m}} =  (25)

∑_{k=1}^{m−1} Cycle_{m+1}({g_{0} : ... : g_{k−1} : g_{k}|u_{0} : ... : u_{k−1} : u_{m}}∧{g_{k} : ... : g_{m}|u_{k} : ... : u_{m}})

Proof. To compute (2.2) we rewrite formula (24) using the transformation formula (17) and substituting u_{k} := t_{0} + ... + t_{k}. We get

δ{g_{0} : ... : g_{m}|u_{0} : ... : u_{m}} =  (26)

∑_{k=1}^{m−1} Cycle_{m+1}({g_{0} : ... : g_{k−1} : g_{k}|u_{0} : ... : u_{k−1} : 0}∧{g_{k} : ... : g_{m}|u_{0} : t_{k+1} : ... : t_{m}})

Since u_{k+i} = u_{k} + t_{k+1} + ... + t_{k+i} we have using (16)

{g_{k} : ... : g_{m}|0 : t_{k+1} : ... : t_{m}} = {g_{k} : ... : g_{m}|u_{k} : u_{k+1} : ... : u_{m}} Notice that u_{m} = 0. The lemma follows.

2. Another way to spell the definition of D_{•,•}(G). Define the elements

I_{n_{0},..,n_{m}}(g_{0} : ... : g_{m}) ∈ D_{w,m}(G) where w := n_{0} + ... + n_{m} − 1 by setting I_{1,n_{1},..,n_{m}}(g_{0} : ... : g_{m}) = {g_{0} : ... : g_{m}}_{n_{1},...,n_{m}} and, more generally

I_{n_{0},n_{1},...n_{m}}(g_{0} : ... : g_{m}) :=  (27)

(−1)^{n_{0}−1} · ∑_{i_{1}+...+i_{m}=n_{0}−1} \binom{n_{1} + i_{1}}{i_{1}}...\binom{n_{m} + i_{m}}{i_{m}}\{g_{0} : ... : g_{m}\}_{n_{1}+i_{1},...,n_{m}+i_{m}}

Here i_{k} ≥ 0. This definition is equivalent to the formula

∑_{n_{i}≥1} I_{n_{0},...,n_{m}}(g_{0} : ... : g_{m})t_{0}^{n_{0}−1} · ... · t_{m}^{n_{m}−1} = {g_{0} : ... : g_{m}|t_{0} : ... : t_{m}} (28)

where the right hand side was defined in (15). In particular we have the cyclic symmetry relations

I_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m}) = I_{n_{1},...,n_{m},n_{0}}(g_{1} : ... : g_{m} : g_{0})  (29)

Remark. For G = μ_{N} the properties of the generators I_{n_{0},n_{1},...,n_{m}}(a_{0} : ... : a_{m}) reflect the properties of the iterated integral

∫_{0}^{a_{0}} \underbrace{\frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n_{0}−1} ∘ \underbrace{\frac{dt}{a_{1} − t} ∘ \frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n_{1}} ∘... ∘ \underbrace{\frac{dt}{a_{m} − t} ∘ \frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n_{m}}  (30)

considered modulo the products of similar integrals and modulo the lower depth integrals. This integral is divergent if n_{0} > 1, and so has to be regularized (see theorem 7.1 in [G1] or [G2]) . Its regularized value is given by formula (27). The shuffle relations (21) correspond precisely to the shuffle product formula for the iterated integrals (30) with n_{0} = 1.

It follows from formula (28) and lemma 2.2 that the coproduct of these elements is given by the following formula:

δI_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m}) =  (31)

Cycle_{m+1}( ∑_{j=1}^{m−1} ∑_{n′_{j}+n′′_{j} =n_{j}+1} I_{n_{0},...,n_{k−1},n′_{k}}(g_{0} : ... : g_{k})∧I_{n_{k},...,n_{m−1},n′′_{m}}(g_{k} : ... : g_{m}))

Notice that the second summation is over positive integers n′_{j}, nj^{′′} such that

(n′_{j} − 1) + (nj^{′′} − 1) = n_{j} − 1.

3. Two related bigraded Lie coalgebras: D̃_{•,•}(G) and D′_{•,•}(G).

i) The generators of D̃_{w,m}(G) are symbols

Ĩ_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m}) ∑ n_{i} = w − 1

The relations are the following: the homogeneuity

Ĩ_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m}) = Ĩ_{n_{0},n_{1},...,n_{m}}(hg_{0} : ... : hg_{m}), h ∈ G (32)

the cyclic symmetry (29), and in addition Ĩ_{1,...,1}(e : ... : e) = 0. It is easy to se that formula (31) provides a bigraded Lie coalgebra structure on D̃_{•,•}(G).

For an associative ℚ-algebra A let C(A) := A/[A, A] be the ℚ-vector space of cyclic words in A. Denote by C(A_{1}A_{2}A_{3}...) the cyclic word A_{1}A_{2}A_{3}... = A_{2}A_{3}...A_{1}.

Let A(G) be the free associative algebra with the generators Y and X_{g}, g ∈ G. The group G acts naturally on the generators of A(G) (as in s. 1.4). So it acts by automorphisms of the algebra itself. Let C̃(A(G)) be the quotient of C(A(G)) by the subspace generated by Y^{n}, X_{g}^{n}, n ≥ 0.

Lemma 2.3 There is a canonical isomorphism of ℚ-vector spaces

η : D̃_{•,•}(G) ⟶ C̃(A(G))_{G}
Ĩ_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m}) ⟼ C (X_{g_{0}}Y^{n_{0}−1}X_{g_{1}}X^{n_{1}−1}...X_{g_{m}}Y^{n_{m}−1}) (33)

Remark. This formula reflects, in the case G = μ_{N}, the structure of the iterated integral (30). Notice the special role of X_{g_{0}}: it corresponds to the integration path from 0 to g_{0}.

Proof. If a cyclic word in X_{g}, Y has a nonzero image in the quotient then it has at least one X_{g}. Using the cyclic symmetry we can move it to the begining of the word. Then it is in the image of η. The injectivity is clear. The lemma is proved.

ii) There is a commutative shuffle product ∘_{Sh} on A(G):

(Z_{1}...Z_{k}) ∘_{Sh} (Z_{k+1}...Z_{k+l}) := ∑_{σ∈Σ_{k,l}} Z_{σ(1)}...Z_{σ(k+l)}

We define a shuffle relation in D̃_{•,•}(G) as the image of an element

X_{e} · {(Y^{n_{0}−1}X_{g_{1}}Y^{n_{1}−1}...X_{g_{k}}Y^{n_{k}−1}) ∘_{Sh} (Y^{m_{0}−1}X_{h_{1}}Y^{m_{1}−1}...X_{h_{l}}Y^{m_{l}−1})}

(34)

where k, l ≥ 0 and n_{0}, m_{0} ≥ 1, under the isomorphism η^{−1}. We define the Lie coalgebra D′_{•,•}(G) as the quotient of D̃_{•,•}(G) by the subspace generated by these shuffle relations.

iii) Let Dw,m^{′′}(G) be the ℚ-vector space generated by the symbols (14) subject to the relations (18),(19), (21) and v). So it has the same generators as D_{w,m}(G), but the relations are relaxed.

Proposition 2.4 There is an isomorphism i : Dw,m^{′′}(G) ⟶ D′_{w,m}(G) given by

i : {g_{0} : g_{1} : ... : g_{m}}_{n_{1},...,n_{m}} ⟼ Ĩ_{1,n_{1},...,n_{m}}(g_{0} : g_{1} : ... : g_{m})  (35)

Proof. Surjectivity of the map i. Let S_{1} be the subspace of A(G)_{G} generated by elements (34) with k = 0 and m_{0} = 1. We write a =^{S_{1}} b if a − b ∈ S_{1}.

One has Ĩ_{n}(g_{0}) = 0. Indeed by the homogeneuty we need to check this only for g_{0} = e. If n = 1 this is relation v). Otherwise \binom{n}{k}X_{e}Y^{n} = X_{e} \{Y^{k} ∘_{Sh} Y^{n−k}\} ∈ S_{1}. Applying η^{−1} we get Ĩ_{n}(e) = 0.

Let us prove by induction on n_{0} the following result, which allows to express Ĩ_{n_{0},n_{1},...,n_{m}}(g_{0} : g_{1} : ... : g_{m}) via Ĩ_{1,n_{1},...,n_{m}}(g_{0} : g_{1} : ... : g_{m}):

Lemma 2.5 a) One has the following formula in A(G):

X_{e}Y^{n_{0}−1}X_{g_{1}}Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1} =^{S_{1}}  (36)

(−1)^{n_{0}−1}X_{e}X_{g_{1}} {(Y^{n_{0}−1}) ∘_{Sh} (Y^{n_{1}−1}X_{g_{2}}Y^{n_{2}−1} · ... · X^{g_{m}}Y^{n_{m}−1})}

b) The shuffle relations from the subspace S_{1} are equivalent to formulae (36).

Remark. Applying η^{−1} to formula (36) we get relations (27).

Proof. When n_{0} = 1 we have nothing to check. In general

0 =^{S_{1}} X_{e} {Y^{k} ∘_{Sh} (X_{g_{1}}Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1})} =  (37)
∑_{j=0}^{k} X_{e}Y^{k−j}X_{g_{1}} {Y^{j} ∘_{Sh} (Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1})}  (38)

By the induction assumption this is equal modulo S_{1} to

X_{e}Y^{k}X_{g_{1}}Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1}+
(−1)^{k−j} ∑_{j=1}^{k} X_{e}X_{g_{1}} {Y^{k−j} ∘_{Sh} Y^{j} ∘_{Sh} (Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1})}  (39)

Therefore X_{e}Y^{k}X_{g_{1}}Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1} equals modulo S_{1} to

(−1)^{k−1} ∑_{j=1}^{k}(−1)^{j}X_{e}X_{g_{1}} \{\binom{k}{j}Y^{k} ∘_{Sh} (Y^{n_{1}−1} · ... · X_{g_{m}}Y^{n_{m}−1})\}  (40)

Since ∑_{j=1}^{k}(−1)^{j}\binom{k}{j} = −1 the statement follows.

Injectivity of the map i. We are dealing with the following problem. One can show that shuffle relations (21) go under the map i precisely to the space

η^{−1}(the subspace of the shuffle relations (34) with n_{0} = m_{0} = 1 )  (41)

(In fact in [G1] relations (21) appeared as a way to write the shufle relations (34) with n_{0} = m_{0} = 1). Let S be the subspace of A(G)_{G} generated by S_{1} and the shuffle relations (34) with n_{0} = m_{0} = 1. We need to show that the general shuffle relations (34) belongs to S. We use induction on n_{0} + m_{0}. One has

Y^{n_{0}−1}X_{g_{1}}Y^{n_{1}−1}X_{g_{2}} · ... · X_{g_{k}}Y^{n_{k}−1} =
Y^{n_{0}−1} ∘_{Sh} X_{g_{1}}Y^{n_{1}−1} · ... · X_{g_{k}}Y^{n_{k}−1} − ∑_{i<n_{0}−1} Y^{i}X_{g_{1}}A_{i}  (42)

Let us take the shuffle product of (42) with the similar expression for Y^{m_{0}−1}X_{h_{1}} · ... · Y^{m_{l}−1}X_{h_{l}}. Notice that

Y^{n_{0}−1} ∘_{Sh} A ∘_{Sh} Y^{m_{0}−1} ∘_{Sh} B = \binom{n_{0} + m_{0} − 2}{n_{0} − 1}Y^{n_{0}+m_{0}−2} ∘_{Sh} A ∘_{Sh} B ∈ S_{1}

and the other terms in the product already belong to the subcpace S by induction. The proposition is proved.

Proposition 2.6 The formula (31) provides a well defined cobracket on D′_{•,•}(G).

Proof. We need only to check that the shuffle relations (34) form a coideal in D̃_{w,m}(G). This is easy to do directly. Another proof see in the end of s. 4.

iv) The Lie coalgebra D_{•,•}(G) is a quotient of D′_{•,•}(G).

3 The Lie algebras of special derivations

1. The Lie algebra of special derivations and the depth filtration. Let 𝕃(X_{0}, ..., X_{n}) be the pronilpotent completion of the free Lie algebra generated by elements X_{0}, ..., X_{n}. Its derivation D is called special if there are S_{i} ∈ 𝕃(X_{0}, ..., X_{n}) such that

D(X_{i}) = [S_{i}, X_{i}], and D(X_{0} + ... + X_{n}) = 0  (43)

Denote by Der^{S}𝕃(X_{0}, ..., X_{n}) the Lie algebra of special derivations of the Lie algebra 𝕃(X_{0}, ..., X_{n}).

The weight filtration on the Lie algebra 𝕃(X_{0}, ..., X_{n}) as an increasing filtration by the lower central series. It goes from −∞ to −1; the weight of X_{i} is −1. It provides the weight filtration on the Lie algebra Der^{S}𝕃(X_{0}, ..., X_{n}).

Let I be the kernel of the natural projection 𝕃(X_{0}, X_{1}, ..., X_{n}) ⟶ ℚ given by X_{i} ⟼ δ_{i0}. Its powers [[[I, I], I], ..., I] provide the depth filtration on 𝕃(X_{0}, ..., X_{n}). It induces the depth filtration on the Lie algebra Der^{S}𝕃(X_{0}, ..., X_{n}). Namely, a derivation (43) is of depth −m if S_{0} is of depth −m, i.e. there are at least m X_{0}’s in S_{0}.

Denote by GrDer_{•,•}^{S}𝕃(X_{0}, ..., X_{n}) the associated graded quotient of the Lie algebra Der^{S}𝕃(X_{0}, ..., X_{n}) with respect to the depth filtration. It is a projective limit of finite dimensional Lie algebras bigraded by the weight and depth.

Let 𝔸(X_{0}, ..., X_{n}) be the completion of the free associative algebra generated by X_{0}, ..., X_{n}. By definition the Lie algebra Der^{S}𝔸(X_{0}, X_{1}, ..., X_{n}) of its special derivations consists of the derivations for which (43) holds with S_{i} ∈ 𝔸(X_{0}, ..., X_{n}). The depth filtration on 𝔸(X_{0}, ..., X_{n}) and Der^{S}𝔸(X_{0}, ..., X_{n}) are defined in a similar way. Let GrDer_{•,•}^{S}𝔸(X_{0}, ..., X_{n}) be the corresponding associated graded Lie algebra.

2. Cyclic words and special derivations ([Dr], [K]). Consider a map of ℚ-linear spaces ∂_{X_{i}} : C(𝔸(X_{0}, X_{1}, ..., X_{n})) ⟶ 𝔸(X_{0}, X_{1}, ..., X_{n}) given on the generators by the following formula (the indices are modulo m):

∂_{X_{j}}C(X_{i_{1}}...X_{i_{m}}) := ∑_{X_{i_{k}} =X_{j}} X_{i_{k+1}}...X_{i_{k+m−1}}

For example ∂_{X_{1}}C(X_{1}X_{2}X_{1}X_{2}^{2}) = X_{2}X_{1}X_{2}^{2} + X_{2}^{2}X_{1}X_{2}. There is a map

κ : C(𝔸(X_{0}, X_{1}, ..., X_{n})) ⟶ Der^{S}𝔸(X_{0}, X_{1}, ..., X_{n})
κC(X_{i_{1}}...X_{i_{m}})(X_{j}) := [∂_{X_{j}}(X_{i_{1}}...X_{i_{m}}), X_{j}]

Here on the right [A, B] = AB − BA. We obviously get a special derivation, i.e.

∑_{j=0}^{n} [∂_{X_{j}}C(X_{i_{1}}...X_{i_{m}}), X_{j}] = 0  (44)

Indeed, for each of the 2m monomial terms in (44) there exists exactly one term in (44) which cancells it. For example X_{i_{1}}...X_{i_{m}} appears, with the opposite signs, in [∂_{X_{i_{1}}} C(X_{i_{1}}...X_{i_{m}}), X_{i_{1}}] and [∂_{X_{i_{2}}} C(X_{i_{1}}...X_{i_{m}}), X_{i_{2}}]

There is a Lie algebra structure on C(𝔸(X_{0}, X_{1}, ..., X_{n})) for which κ is a Lie algebra morphism. Namely, the space of cyclic words C(𝔸(X_{0}, ..., X_{n})) has a natural associative algebra structure given by

C(Y_{1} · ... · Y_{k}) ∗ C(Z_{1} · ... · Z_{l}) := ∑_{Y_{i}=Z_{j}} C(Y_{i+1} · ... · Y_{i} · Z_{j+1} · ... · Z_{j−1}) (45)

where Y_{α}, Z_{β} are the generators X_{i}, and the summation is over all pairs Y_{i} = Z_{j}. It is easy to check that it is associative. So there is a Lie algebra structure

[C(Y_{1} · ... · Y_{k}), C(Z_{1} · ... · Z_{l})] :=

\frac{1}{2} ∑_{Y_{i}=Z_{j}}(C(Y_{i+1} · ... · Y_{i} · Z_{j+1} · ... · Z_{j−1}) − C(Z_{j+1} · ... · Z_{j−1} · Z_{j} · Y_{i+1} · ... · Y_{i−1}))

(We multiply by \frac{1}{2} the usual formula for [,]). Denote by ∗_{1} the induced product on the associated graded quotient GrC(𝔸(X_{0}, ..., X_{n})) with respect to the depth filtration. It is given by fomula (45) where the summation is over Y_{i} = Z_{j} different from X_{0} only.

The centralizer of X_{i} in 𝔸(X_{0}, ..., X_{n}) is ℚ[[X_{i}]]. So Kerκ = ⊕_{i=0}^{n}ℚ[[X_{i}]]. Let C̃(𝔸(X_{0}, ..., X_{n})) := C(A(X_{0}, ..., X_{n}))/ ⊕_{i=0}^{n} ℚ[[X_{i}]].

Proposition 3.1 κ : C̃(𝔸(X_{0}, ..., X_{n})) ⟶ Der^{S}(𝔸(X_{0}, ..., X_{n})) is a Lie algebra isomorphism.

Proof. We need only to show that κ is surjective. This follows from theorem 4.2 in [K]. For the convinience of the reader we reproduce the argument. Set A := 𝔸(X_{0}, ..., X_{n}). Let F^{1}(A) := ⊕_{i=0}^{n}A ⊗ dX_{i}. There is a sequence

0 ⟶ C(A) = A/[A, A] ⟶^{d} F^{1}(A) ⟶^{t} [A, A] ⟶ 0  (46)

where d(C) := ∑ ∂_{X_{i}}C ⊗ dX_{i} and t[Z ⊗ dX_{i}] := [Z, X_{i}]. It is a complex by (44), and it is clearly exact from the left and right. There is an isomorphism of filtered (by the weight filtration) ℚ-vector spaces A ⟶ F^{1}(A), X_{i_{1}}...X_{i_{m}} ⟶ X_{i_{1}}...X_{i_{m−1}} ⊗ dX_{i_{m}}. Thus the Euler characteristic of the weight ≥ −m part of (46) is zero. So the complex is exact. The proposition proved.

3. Special equivariant derivations. Let 𝔸(G) be the completion of the tensor algebra generated by X_{g}, g ∈ G, and Y. Let C̃(𝔸(G))_{G} be the space of coinvariants of G. Let us assume that G is finite. We define a linear map

κ_{G} : C̃(𝔸(G))_{G} ⟶ Der^{S}𝔸(G)^{G} =: Der^{SE}𝔸(G)

as a composition of κ with the projection to G-invariants:

C(X_{g_{0}}Y^{n_{0}−1} · ... · X_{g_{m}}Y^{n_{m}−1})_{G} ⟼ \frac{1}{|G|} ∑_{h∈G} κ(C(X_{hg_{0}}Y^{n_{0}−1} · ... · X_{hg_{m}}Y^{n_{m}−1}))

here C(...)_{G} means the image of a cyclic word C(...) in C̃(𝔸(G))_{G}. Proposition 3.1 immediately implies

Proposition 3.2 κ_{G} is a Lie algebras isomorphism.

The algebra structure on the associate graded quotient GrC̃(𝔸(G))_{G} with respect to the depth filtration is described as follows (see the picture)

C(X_{g_{0}}Y^{n_{0}−1} · ... · X_{g_{k}}Y^{n_{k}−1})_{G} ∗_{1} C(X_{h_{0}}Y^{m_{0}−1} · ... · X_{h_{l}}Y^{m_{l}−1})_{G} =

∑_{i,j} C(X_{g_{i}^{−1}g_{i+1}}Y^{n_{i+1}−1}·...·X_{g_{i}^{−1}g_{i}}Y^{n_{j}−1}X_{h_{j}^{−1}h_{j+1}}Y^{m_{j+1}−1}·...·X_{h_{j}^{−1}h_{j−1}}Y^{m_{j−1}−1})_{G}

It can be defined by the same formulae for any abelian group G.

4 The dihedral Lie algebras and special equivariant derivations

1. Assume that G is a finite group. Consider the following formal expression:

ξ_{G} = ∑_{cyclic words} Ĩ_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m})⊗C(X_{g_{0}}Y^{n_{0}−1}·...·X_{g_{m}}Y^{n_{m}−1})_{G} (47)

where the sum is over all cyclic words in X_{g}, Y. (This is not the same as the sum over n_{i} ≥ 1). Applying the map Id ⊗ κ_{G} we get a bidegree (0, 0) element

ξ̃_{G} ∈ D̃_{•,•}(G)ˆ⊗_{ℚ}GrDer_{•,•}^{SE}𝔸(G)

Notice that D_{•,•}(G) is an inductive limit of bigraded Lie coalgebras, which are finite dimensional since G is a finite group. So its dual D_{•,•}(G) is a projective limit of bigraded Lie algebras, and 𝔻_{−w,−m}(G) = D_{w,m}(G)^{∨}.

We may view the element ξ_{G} as a map between the bigraded ℚ-vector spaces:

ξ_{G} ∈ Hom_{ℚ−V ect}(𝔻_{•,•}(G), GrDer_{•,•}^{SE}𝔸(G))  (48)

Notice that 𝔻̃_{•,•}(G) ⊂ 𝔻′_{•,•}(G) ⊂ 𝔻_{•,•}(G).

Theorem 4.1 a) ξ̃_{G} : 𝔻̃_{•,•}(G) ⟶^{=} GrDer_{•,•}^{SE}𝔸(G) is an isomorphism of the bigraded Lie algebras.

b) Restricting ξ̃_{G} to 𝔻′_{•,•}(G) we get an isomorphism ξ′_{G} : 𝔻′_{•,•}(G) ⟶^{=} GrDer_{•,•}^{SE}𝕃(G).

c) Restricting ξ′_{G} to 𝔻_{•,•}(G) we get an injective Lie algebra morphism.

Proof. a) We start with a remark from linear algebra. Let L_{1}, L_{2} be Lie algebras and ζ ∈ Hom_{ℚ−V ect}(L_{1}, L_{2}) = L_{1}^{∗} ⊗ L_{2}. Denote by δ : L_{1}^{∗} ⟶ Λ^{2}L_{1}^{∗} the Lie cobracket on L_{1}^{∗} and by [,] : Λ^{2}L_{2} ⟶ L_{2} the Lie bracket on L_{2}. Notice that S^{2}(L_{1}^{∗} ⊗L_{2}) = Λ^{2}L_{1}^{∗} ⊗Λ^{2}L_{2} ⊕S^{2}L_{1}^{∗} ⊗S^{2}L_{2}. Denote by ζ ∘ζ ∈ Λ^{2}L_{1}^{∗} ⊗Λ^{2}L_{2} the first component of the symmetric square of ζ. Then ζ is a morphism of Lie algebras if and only if

(δ ⊗ id)(ζ) = (id ⊗ [,])(ζ ∘ ζ)

If ζ = ∑_{i} A_{i} ⊗ B_{i} then (id ⊗ [,])(ζ ∘ ζ) = ∑_{i,j} A_{i} ∧ A_{j} ⊗ [B_{i}, B_{j}] and so the condition is

∑_{i} δ(A_{i}) ⊗ B_{i} = ∑_{i,j} A_{i} ∧ A_{j} ⊗ [B_{i}, B_{j}]  (49)

Let us return to our situation. To visualize a single expression

δ(Ĩ_{n_{0},n_{1},...,n_{m}})(g_{0} : ... : g_{m}) ⊗ C(X_{g_{0}}Y^{n_{0}−1} · ... · X_{g_{m}}Y^{n_{m}−1})_{G}  (50)

in (δ ⊗ id)(ξ_{G}) we proceed as follows. Take an oriented circle divided on m + 1 arcs by m + 1 black points. Label each of the black points by an element of the set {X_{g}}, g ∈ G and call it an X_{g}-point. The i-th arc is subdivided on n_{i} little arcs by n_{i} − 1 points labeled by Y (presented by little circles on the picture and called Y -points). Call such a picture a circle with (n_{0}, ..., n_{m})-cyclic G-structure.

It corresponds to a cyclic word C(X_{g_{0}}Y^{n_{0}−1} · ... · X_{g_{m}}Y^{n_{m}−1})_{G} in an obvious way: put X_{g} for X_{g}-points and Y for Y -points and make a cyclic word.

The group G acts on circles with (n_{0}, ..., n_{m})-cyclic G-structure: an element h ∈ G transforms X_{g}-points to X_{hg}-points and leaves untouched Y -points. The coinvariants are called circles with (n_{0}, ..., n_{m})-cyclic equivariant G-structure. They correspond to expressions (50) as well as to the generators Ĩ_{n_{0},n_{1},...,n_{m}}(g_{0} : ... : g_{m}) obeying the cyclic symmetry and homogeneuty condition (32).

Let us mark such a picture by choosing in addition a little arc and a black point different from the ends of the arc containing the little arc. We call it a marked circle with (n_{0}, ..., n_{m})-cyclic equivariant G-structure.

It follows from fromula (31) that expresion (50) is a sum of the terms which are in bijective correspondence with the marked circles with (n_{0}, ..., n_{m})-cyclic equivariant G-structure. For example the marked circle with (3, 2, 2, 1)-cyclic structure on the picture corresponds to I_{2,1} ∧ I_{2,1,2} ⊗ C(XY^{2}XY XY^{3}XY^{2}). In general we use first the marks to cut the circle on 2 oriented semicircles and make 2 new circles by gluing the endpoints of the semicircles, adding a new black point on each of the new circles instead of the marked black point on the initial circle, and using the rest of the points on each of the new circles. The new circles are getting natural cyclic structures.

On the other hand the terms of the sum

∑ Ĩ_{p_{0},...,p_{k}}(g_{0} : ... : g_{k}) ∧ Ĩ_{q_{0},...,q_{l}}(h_{0} : ... : h_{l})⊗

C(X_{g_{0}}Y^{p_{0}−1} · ... · X_{g_{k}}Y^{p_{k}−1})_{G} ∗_{1} C(X_{h_{0}}Y^{q_{0}−1} · ... · X_{h_{l}}Y^{q_{l}−1})_{G}

where the summation is over pars of cyclic words C(X_{g_{0}}Y^{p_{0}−1} · ... · X_{g_{k}}Y^{p_{k}−1})_{G} and C(X_{h_{0}}Y^{q_{0}−1} · ... · X_{h_{l}}Y^{q_{l}−1})_{G} and ∗_{1} is the product in GrC(𝔸(G))_{G}, also correspond to the marked circles with cyclic equivariant G-structure. Indeed, the product

C(X_{g_{0}}Y^{p_{0}−1} · ... · X_{g_{k}}Y^{p_{k}−1})_{G} ∗_{1} C(X_{h_{0}}Y^{q_{0}−1} · ... · X_{h_{l}}Y^{q_{l}−1})_{G}

of P := (p_{0}, ..., p_{k}) and Q := (q_{0}, ..., q_{l})-cyclic words is a sum of (k + 1)(l + 1) cyclic words. Let us picture P and Q-cyclic words by the corresponding circles with P and Q-cyclic G-structures. Then each of the (k + 1)(l + 1) terms in the product corresponds to a pair {a circle with P -cyclic G-structure + a black point on it, a circle with Q-cyclic G-structure + a black point on it}.

Such pairs are in bijective correspondence with marked circles with cyclic equivariant G-structures, as demonstrated on the picture:

Namely, to get a marked circle with cyclic equivariant G-structure we proceed as follows. Make a connected sum of the P and Q-circles by connecting the choosen black points by a bridge. The orientation of the initial circles induces an orientation of their connected sum. Instead of the choosen two black points on the initial circles we put a single black point on the bottom part of the bridge. We keep the rest of the points, getting a cyclic structure on the new circle. The black point on the bottom bridge together with the top part of the bridge provide the marks on the circle with the cyclic structure we get. The distinguished point on each of the three circles is choosen to be the X_{e}-point. The procedure is clearly reversable, so we get the promised bijection.

We proved that ξ̃_{G} is a morphism of bigraded Lie algebras. By lemma 2.3 it is an isomorphism. The part a) of the theorem is proved.

Lemma 4.2 Let D ∈ Der^{SE}𝔸(G). Then D(X_{e}) ∈ 𝕃(G) <=> D ∈ Der^{SE}𝕃(G).

Proof. Since D is equivariant D(X_{e}) ∈ 𝕃(G) implies that D(X_{g}) ∈ 𝕃(G) for any g ∈ G. Since D(∑ X_{g} + Y) = 0 we get also D(Y) ∈ 𝕃(G).

The algebra 𝔸(X_{0}, ..., X_{n}) is a commutative Hopf algebra with the coproduct Δ(X_{i_{1}}...X_{i_{m}}) := ∑ X_{i_{α_{1}}} ...X_{i_{α_{k}}} ⊗X_{i_{β_{1}}} ...X_{i_{β_{l}}} where the sum is over all partitions of {i_{1}, ..., i_{n}} on two subsets 1 ≤ α_{1} < ... < α_{k} ≤ m and 1 ≤ β_{1} < ... < β_{k} ≤ m. The Hopf algebra A(X_{0}, ..., X_{n}) with the shufle product ∘_{Sh} is its dual. Let Δ̃(Z) := Δ − Id ⊗ Z − Z ⊗ Id. Then for D ∈ ξ′_{G}(𝔻_{•,•}(G)) one has

D(X_{e}) ∈ 𝕃(G) <=> (id ⊗ Δ̃ ∘ ∂_{X_{e}})(ξ′_{G}) ∈ D_{•,•}(G)ˆ⊗(𝔸(G) ⊗ 𝔸(G)) is zero.

The element A⊗B ∈ 𝔸(G)⊗𝔸(G) appears in (id⊗Δ̃∘∂_{X_{e}})(ξ′_{G}) with coefficient X_{e}(A ∘_{Sh} B). Therefore shuffle relations (34) are equivalent to the condition ξ̃_{G}(D_{•,•}(G)) ⊂ GrDer_{•,•}^{SE}𝕃(G). The lemma is proved.

Proof of proposition 2.6. We just proved that ξ′_{G} is an isomorphism of bigraded ℚ-vector spaces. Using a) (and an obvious fact that Der^{SE}𝕃(G) is a Lie algebra!) we conclude that 𝔻′_{•,•}(G) is a Lie subalgebra of 𝔻̃_{•,•}(G).

So ξ′_{G} is a Lie algebra isomorphism. The part b) of the theorem is proved.

c) It follows from a), b) and propositions 2.4 and 2.6. The theorem is proved.

5. Towards conjecture 1.2: the distribution relations. Let X_{N} := ℙ^{1}\0, μ_{N}, ∞). There are maps

m_{N} : X_{NM} ⟶ X_{M}, z ⟼ z^{N}, i_{N} : X_{NM} ↪ X_{M}, i_{N} : z ⟼ z

They induce the homomorphisms

i_{N∗} : 𝕃(μ_{MN}) ⟶ 𝕃(μ_{M}); Y ⟶ Y, X_{ζ} ⟶ \{ \begin{matrix}0 & ζ ∉ μ_{M} \\ ζ & ζ ∈ μ_{M} \end{matrix}

m_{N∗} : 𝕃(μ_{MN}) ⟶ 𝕃(μ_{M}); Y ⟶ NY, X_{ζ} ⟶ X_{ζ^{N}}

So they induce the maps m̃_{N∗}, ĩ_{N∗} : Der_{•,•}^{SE}𝕃(μ_{NM}) ⟶ Der_{•,•}^{SE}𝕃(μ_{N}). We claim that G^{(l)}(μ_{NM}) ⊂ Ker(ĩ_{N∗} − m̃_{N∗}). Indeed, each of the maps i_{N}, m_{N} provides a commutative diagram

Gal(\overline{ℚ}/ℚ(ζ_{l^{∞}}, ζ_{MN})) ⟶^{φ_{MN}^{(l)}} Autπ_{1}^{(l)}(X_{MN})
↓ j_{N}  ↓
Gal(\overline{ℚ}/ℚ(ζ_{l^{∞}}))  ⟶^{φ_{N}^{(l)}} Autπ_{1}^{(l)}(X_{N})

where j_{N} is the natural inclusion. So the composition φ_{N}^{(l)} ∘ j_{N} does not depend on the choise of the maps i_{N}, m_{N}. Thus an element of Der_{•,•}^{SE}𝕃(μ_{MN}) given by

∑_{n_{i}>0} a(e : g_{1} : ... : g_{m})_{n_{0},..,n_{m}} · κ_{μ_{MN}} C(X_{e}Y^{n_{0}−1}...X_{g_{m}}Y^{n_{m}−1})

(the sum is over cyclic words) is in Ker(ĩ_{N∗} − m̃_{N∗}) if and only if

a(e : g_{1} : ... : g_{m})_{n_{0},..,n_{m}} = N^{w−m} ∑_{h_{i}^{N} =g_{i}} a(h_{0} : h_{1} : ... : h_{m})_{n_{0},..,n_{m}}

This is precisely the distribution relations.

To complete the proof of conjecture 1.2 it remains to show that G^{(l)}(μ_{N}) lies in the subspace of Der_{•,•}^{SE}𝕃(μ_{N}) defined by the shuffle relations (22). These relations are obviously satisfied by the power series defining the multiple polylogarithms (see formula (1) in [G1]). Unfortunately I do not know any good proof of them via derivations or the iterated integrals (30). There is, however, a very indirect argument: relations (22) hold for functions => for the corresponding framed mixed Hodge structures => for the corresponding mixed Tate motives over ℚ(ζ_{N}) => valid for their l-adic realization. I will publish it elswhere. It is rather amusing that the simplest relations on the level of functions are responsible for the most nontrivial constraints on the image of the Galois group!

5 Proofs of theorems 1.4 and 1.5

Applying to {e : e|t_{0} : t_{1}} = ∑_{n>0}{e : e}_{n}(t_{0} − t_{1})^{n−1} the cyclic symmetry relation we get D_{−2w,−1} = 0. Recall that D_{−1,−1} = 0 by definition.

Since GrG_{•,−1}^{(l)} is abelian we may identify it with the ℚ_{l}-points of the corresponding algebraic group. So it makes sense to talk about projection of φ^{(l)}(σ) on GrG_{•,−1}^{(l)}. Let φ_{−w,−1}^{(l)}(σ) be the component of this projection in GrG_{−w,−1}^{(l)}.

In [So] Soulé constructed for each integer m > 1 a Gal(ℚ(ζ_{l^{∞}})/ℚ)-homomorphism χ_{m} : Gal(\overline{ℚ}/ℚ(ζ_{l^{∞}}))^{ab} ⟶ ℚ_{l}(m) and proved that it is zero if and only if m is even. Let I_{2m−1}(e)^{∨} be the generator of 𝔻_{−2m+1,−1} dual to I_{2m−1}(e). Then it follows from the Key Lemma B in [Ih2] that

φ^{(l)}(σ) = \frac{(1 − l^{m−1})^{−1}}{(m − 1)!} χ_{m}(σ)ξ(I_{2m−1}(e)^{∨}) (m : odd ≥ 3)

So 𝔻_{•,−1} ⊗ ℚ_{l} = L_{•,−1}^{(l)}. Since both 𝔻_{•,•} ⊗ ℚ_{l} and L_{•,•}^{(l)} are Lie subalgebras of Der_{•,•}^{SE}𝕃, and in addition 𝔻_{•,−1} ⊗ ℚ_{l} = L_{•,−1}^{(l)} we have L_{•,•}^{(l)} ⊂ 𝔻_{•,•} ⊗ ℚ_{l}. Thus there are the following commutative diagrams ([,] are the commutator maps):

Λ^{2}L_{•,−1}^{(l)} ⟶^{=} Λ^{2}𝔻_{•,−1} ⊗ ℚ_{l}  L_{•,−2}^{(l)} ⊗ L_{•,−1}^{(l)} ⟶^{=} 𝔻_{•,−2} ⊗ 𝔻_{•,−1} ⊗ ℚ_{l}
↓ [,]  ↓ [,]  and  ↓ [,]  ↓ [,]
L_{•,−2}^{(l)} ↪ 𝔻_{•,−2} ⊗ ℚ_{l}  L_{•,−3}^{(l)}  ↪  𝔻_{•,−3} ⊗ ℚ_{l}

Theorem 5.1 Let H_{(w,m)} be the weight w, depth m part of H. Then

H_{(w,2)}^{i}(𝔻_{•,•}) = H_{cusp}^{i−1}(GL_{2}(ℤ), S^{w−2}V_{2}), i = 1, 2  (51)
H_{(w,3)}^{i}(𝔻_{•,•}) = H_{cusp}^{i}(GL_{3}(ℤ), S^{w−3}V_{3}) = 0, i = 1, 2, 3  (52)

Proof. Let ℚ_{−1,−1} be the one dimensional Lie algebra placed in degree (−1, −1). There is an exact sequence 0 ⟶ 𝔻_{•,•} ⟶ 𝔻̂_{•,•} ⟶ ℚ_{−1,−1} ⟶ 0. It provides the follwing bicomplex where the horizontal lines are the depth −m parts of the standard cochain complex for the Lie algebra 𝔻̂_{•,•}:

𝔻̂_{•,−3} ⟶^{δ} 𝔻̂_{•,−2} ⊗ 𝔻̂_{•,−1} ⟶^{δ} Λ^{3}𝔻̂_{•,−1}
↓ ∂  ↓ ∂
𝔻̂_{•,−2} ⟶^{δ} Λ^{2}𝔻̂_{•,−1}
↓ ∂
𝔻̂_{•,•}

The vertical differentials have bidegree (1, 1) and are constracted as follows. Let V be a vector space and f ∈ V^{∗}. Define a map ∂ : Λ^{n}V ⟶ Λ^{n−1}V, ∂^{2} = 0, by v_{1} ∧ ... ∧ v_{n} ⟶ ∑_{i=1}^{n}(−1)^{i−1}f(v_{i})v_{1} ∧ ... ∧ v̂_{i} ∧ ... ∧ v_{n}. There is a canonical functional 1 ∈ ℚ_{−1,−1}^{∨} ⊂ 𝔻̂_{•,•}. It provides the maps ∂ : Λ^{n}𝔻̂_{•,•} ⟶ Λ^{n−1}𝔻̂_{•,•}.

Consider the inclusion of the subcomplex

𝔻_{•,−2} ⟶ Λ^{2}𝔻_{•,−1} ↪ 𝔻_{•,−2} ⟶ Λ^{2}𝔻̂_{•,−1} ⟶ 𝔻̂_{•,−1}  (53)

into the complex given by the two bottom lines of the bicomplex. The isomorphism μ from theorem 1.2 in [G1] is easyly extended to an isomorphism between the complex 𝕄_{(2)}^{∗} ⊗_{GL_{2}(ℤ)} S^{•−2}V_{2} and the complex (53) which takes τ_{[1,2]}𝕄_{(2)}^{∗} ⊗_{GL_{2}(ℤ)} S^{•−2}V_{2} just to the subcomplex (53). This proves the part a).

The depth −3 row of the bicomplex can be written as

𝔻_{•,−3} ⟶^{δ} 𝔻_{•,−2} ⊗ 𝔻̂_{•,−1} ⟶^{δ} 𝔻_{•,−1} ∧ 𝔻_{•,−1} ∧ 𝔻̂_{•,−1}  (54)

So it is maped by ∂ onto the subcomplex (53) and 𝔻_{•,−3} ⟶ 𝔻_{•,−2} ⊗ 𝔻_{•,−1} ⟶ Λ^{2}𝔻_{•,−1} is the kernel. By theorem 1.3 and formula (36) in [G1] we have

H_{(w,3)}^{i}(𝔻̂_{•,•}) = \{ \begin{matrix}0 & i = 1, 2 \\ H_{cusp}^{1}(GL_{2}(ℤ), S^{w−2}V_{3}) & i = 3 \end{matrix}  (55)

According to the part a) H_{cusp}^{1}(GL_{2}(ℤ), S^{w−2}V_{3}) is the only nonzero cohomology group of subcomplex (53), the map ∂ provides a quasiisomorphism of the complex (54) and the subcomplex (53). Theorem 5.1 is proved.

The left vertical arrows in the commutative diagrams above are surjective by definition. Since H_{cusp}^{0}(GL_{2}(ℤ), S^{w−2}V_{2}) == 0 if w > 2 the right vertical arrows there are also surjective by theorem 5.1. So the bottom horizontal maps are isomorphisms. The dimensions of 𝔻_{−w,−2} and 𝔻_{−w,−3} has been computed in theorems 1.4 and 1.5 in [G1], and are given by the right hand sides of formulas

(10) and (13). This implies both theorems 1.4 and 1.5 These theorems, in particular, imply that the double shuffle relations provide

a complete list of constraints on the Lie algebra of the image of the Galois group in Autπ_{1}^{(l)}(ℙ^{1}\{0, 1, ∞}) in the depths −2 and −3.

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