2.1 The motivic $\pi _1$ of $\mathbb {P}^1\backslash \{0,1,\infty \}$

Let $X=\mathbb {P}^1\backslash \{0,1,\infty \}$, and let $\overset {\rightarrow }{1}\!_0, -\overset {\rightarrow }{1}\!_1$ denote the tangential base points on $X$ given by the tangent vector $1$ at $0$, and the tangent vector $-1$ at $1$. Denote the de Rham realization of the motivic fundamental torsor of paths on $X$ with respect to these basepoints by

\[ {}_0 \Pi_{1}= \pi^{\mathrm{dR}}_1 ( X,\overset{\rightarrow}{1}\!_0, -\overset{\rightarrow}{1}\!_1) . \]

It is the affine scheme over $\mathbb {Q}$ which to any commutative unitary $\mathbb {Q}$-algebra $R$ associates the set of group-like formal power series in two non-commuting variables $e_0$ and $e_1$,

\[ \{ S \in R\langle \!\langle e_0, e_1 \rangle \! \rangle^\times: \Delta S= S \hat{\otimes} S\} , \]

where $\Delta$ is the completed coproduct for which the elements $e_0$ and $e_1$ are primitive: $\Delta e_i = 1\otimes e_i + e_i \otimes 1$ for $i=0,1$. The ring of regular functions on ${}_0 \Pi _{1}$ is the $\mathbb {Q}$-algebra

\[ \mathcal{O}({}_0 \Pi_{1})\cong \mathbb{Q} \langle e^0, e^1 \rangle \]

whose underlying vector space is spanned by the set of words $w$ in the letters $e^0, e^1$, together with the empty word, and whose multiplication is given by the shuffle product $\, \unicode{x0448} \,: \mathbb {Q} \langle e^0, e^1 \rangle \otimes _{\mathbb {Q}} \mathbb {Q} \langle e^0, e^1 \rangle \rightarrow \mathbb {Q} \langle e^0, e^1 \rangle$. The deconcatenation of words defines a coproduct, making $\mathcal {O}({}_0 \Pi _{1})$ into a Hopf algebra. This gives rise to a group structure on ${}_0 \Pi _{1}(\mathbb {Q})$ (corresponding to the fact that in the de Rham realization there is a canonical path between any two points on $X$). Any word $w$ in $e^0,e^1$ defines a function

\[ {}_0 \Pi_{1}(R) \longrightarrow R \]

which extracts the coefficient $S_w$ of the word $w$ (viewed in $e_0,e_1$) in a group-like series $S \in R\langle \!\langle e_0, e_1 \rangle \! \rangle ^{\times }$. The Lie algebra of ${}_0 \Pi _{1}(\mathbb {Q})$ is the completed Lie algebra $\mathbb {L}(e_0,e_1)^\wedge$ of the graded Lie algebra $\mathbb {L}(e_0,e_1)$ which is freely generated by the two elements $e_0,e_1$ in degree $-1$. The universal enveloping algebra $\mathcal {U} \, \mathbb {L}(e_0,e_1)$ of $\mathbb {L}(e_0,e_1)$ is the tensor (co)algebra on $e_0,e_1$:

(2.1)\begin{equation} T(e_0,e_1)= \bigoplus_{n\geq 0} (\mathbb{Q}\, e_0 \oplus \mathbb{Q} \, e_1)^{{\otimes} n} . \end{equation}

It is the graded cocommutative Hopf algebra which is the graded dual of $\mathcal {O}({}_0 \Pi _{1})$. Its multiplication is given by the concatenation product, and its coproduct is the unique coproduct for which $e_0$ and $e_1$ are primitive.

2.2 Action of the motivic Galois group

Now let $\mathcal {MT}(\mathbb {Z})$ denote the Tannakian category of mixed Tate motives over $\mathbb {Z}$, with canonical fiber functor given by the de Rham realization. Let $\mathcal {G}_{\mathcal {MT}(\mathbb {Z})}$ denote the group of automorphisms of this fiber functor. It is an affine group scheme over $\mathbb {Q}$. It has a decomposition as a semi-direct product

\[ \mathcal{G}_{\mathcal{MT}(\mathbb{Z})}\cong \mathcal{U}_{\mathcal{MT}(\mathbb{Z})} \rtimes \mathbb{G}_m , \]

where $\mathcal {U}_{\mathcal {MT}(\mathbb {Z})}$ is pro-unipotent. Furthermore, one knows from the relationship between the Ext groups in $\mathcal {MT}(\mathbb {Z})$ and Borel's results on the rational algebraic $K$-theory of $\mathbb {Q}$ that the graded Lie algebra of $\mathcal {U}_{\mathcal {MT}(\mathbb {Z})}$ is non-canonically isomorphic to the Lie algebra freely generated by one generator $\sigma _{2i+1}$ in degree $-(2i+1)$ for every $i\geq 1$. It is important to note that only the classes of the elements $\sigma _{2i+1}$ in the abelianization $\mathcal {U}_{\mathcal {MT}(\mathbb {Z})}^{\mathrm {ab}}$ are canonical, the elements $\sigma _{2i+1}$ themselves are not.

Since $\mathcal {O}({}_0 \Pi _{1})$ is the de Rham realization of an Ind-object in the category $\mathcal {MT}(\mathbb {Z})$, there is an action of the motivic Galois group $\mathcal {G}_{\mathcal {MT}(\mathbb {Z})}$ on ${}_0 \Pi _{1}$ and hence

(2.2)\begin{equation} \mathcal{U}_{\mathcal{MT}(\mathbb{Z})} \times {}_0 \Pi_{1} \longrightarrow {}_0 \Pi_{1} . \end{equation}

The action of $\mathcal {U}_{\mathcal {MT}(\mathbb {Z})}$ on the unit element $1\in {}_0 \Pi _{1}$ defines a map

(2.3)\begin{equation} g\mapsto g(1): \mathcal{U}_{\mathcal{MT}(\mathbb{Z})} \longrightarrow {}_0 \Pi_{1} , \end{equation}

and the action (2.2) factors through a map

(2.4)\begin{equation} \circ : {}_0 \Pi_{1} \times {}_0 \Pi_{1} \longrightarrow {}_0 \Pi_{1} \end{equation}

first computed by Ihara. It is obtained from [Reference Deligne and GoncharovDG05, §§ 5.9, 5.13], by reversing all words in order to be consistent with our convention for composition of paths. An element $a\in {}_0 \Pi _{1}$ defines an action denoted by $\langle a \rangle _0$ on ${}_0 \Pi _{0}$ which is compatible with (2.4) via the composition of paths ${}_0 \Pi _{0} \times {}_0 \Pi _{1} \rightarrow {}_0 \Pi _{1}$. Therefore, writing $x_{00}$ (respectively, $x_{01}$) for the element $x$ in ${}_0 \Pi _{0}$ (respectively, ${}_0 \Pi _{1})$, one finds that for $g\in {}_0 \Pi _{1}$,

(2.5)\begin{equation} a\circ g = a \circ g_{01} = a \circ (g_{00}. 1_{01}) = \langle a\rangle_0 (g) . a(1_{01})= (\langle a\rangle_0 (g)) . a , \end{equation}

where $\langle a \rangle _0$ acts on the generators $\exp (e_i)$ in ${}_0 \Pi _{0}$, for $i=0,1$, by

(2.6)\begin{equation} \begin{aligned} \langle a \rangle_0 ( \exp(e_0)) &= \exp(e_0), \\ \langle a \rangle_0 (\exp(e_1)) &= a \exp(e_1) a^{{-}1}.\end{aligned} \end{equation}

We now give a concrete way to compute $\circ$ by expressing it as the restriction of a combinatorially defined map $\, \underline {\circ }\,$ on a larger space.

Definition 2.1 Inductively define a $\mathbb {Q}$-bilinear map

\[ \, \underline{\circ}\, : T(e_0,e_1) \otimes T(e_0,e_1) \longrightarrow T(e_0,e_1) \]

as follows. For any two words $a, w$ in $e_0,e_1$, and any integer $n\geq 0$, let

(2.7)\begin{equation} a \, \underline{\circ}\, (e_0^{n} e_1 w) = e_0^n a e_1 w + e_0^n e_1 a^* w +e_0^n e_1 (a \, \underline{\circ}\, w), \end{equation}

where $a \, \underline {\circ }\, e_0^n = e_0^n a$, and for any $a_i \in \{e_0,e_1\}$, $(a_1\ldots a_n)^* = (-1)^n a_n \ldots a_1$.

We now show that the map $\, \underline {\circ }\,$ restricts to the full action

(2.8)\begin{equation} \mathfrak{g} \otimes_{\mathbb{Q}} \mathcal{U} \, \mathrm{Lie}\,{}_0 \Pi_{1} \rightarrow \mathcal{U} \, \mathrm{Lie}\,{}_0 \Pi_{1} \end{equation}

induced by (2.4). To see this, identify the universal enveloping of the graded Lie algebra $\mathcal {U} \,\mathrm {Lie} \, {}_0 \Pi _{1} = \mathcal {U} \mathbb {L}(e_0,e_1)$ with $T(e_0, e_1)$. Since $\mathfrak {g}$ is isomorphic (as a vector space) to $\mathbb {L}(e_0,e_1)$, there is also a natural embedding $i: \mathfrak {g}\rightarrow T(e_0,e_1)$.

Proposition 2.2 The action (2.8) is obtained by restriction of $\, \underline {\circ }\,$, that is,

\[ a \circ b = i(a) \, \underline{\circ}\, b . \]

Proof. For any $S \in {}_0 \Pi _{1}$, the coefficient of $w$ in $S^{-1}$ is equal to the coefficient of $w^*$ in $S$, since the map $*$ is the antipode in $\mathcal {O}({}_0 \Pi _{1})$. Identifying the (vector space) $\mathfrak {g}\cong \mathbb {L}(e_0,e_1)$ with its image in $T(e_0,e_1)$, it follows that the infinitesimal, weight-graded, version of the map (2.6) is the derivation $\langle f\rangle _0:\mathbb {L}(e_0,e_1) \rightarrow \mathbb {L}(e_0,e_1)$ which for any $f\in \mathbb {L}(e_0,e_1)$ is given by

\[ e_0\mapsto 0 , \quad e_1 \mapsto f e_1 + e_1 f^* . \]

Thus $\langle f\rangle _0 : T(e_0,e_1) \rightarrow T(e_0,e_1)$ is the map

\begin{align*} \langle f \rangle_0 \, (e_0^{m_0} e_1 \ldots e_1 e_0^{m_r} ) &= e_0^{m_0}(f e_1 +e_1 f^*) e_0^{m_1} e_1 \cdots e_0^{m_{r-1}} e_1 e_0^{m_r} \\ &\quad +\cdots + e_0^{m_0} e_1 e_0^{m_1} \cdots e_0^{m_{r-1}}(f e_1+ e_1 f^*) e_0^{m_r} . \end{align*}

Adding the term $e_0^{m_0} e_1 \ldots e_1 e_0^{m_r} f$ corresponding to concatenation on the right by $f$ gives precisely the map defined by (2.7).

2.3 The motivic Lie algebra

By (2.3), we obtain a map of Lie algebras

(2.9)\begin{equation} \mathrm{Lie} (\mathcal{U}_{\mathcal{MT}(\mathbb{Z})}) \longrightarrow \mathfrak{g} = \big(\mathbb{L}(e_0,e_1) \ , \ \{ \,{,}\,\} \big) , \end{equation}

where the Ihara bracket satisfies $\{f,g\}= f\circ g- g \circ f$. It follows from Theorem 1.1 that this map is injective [Reference BrownBro12]. In this paper, we shall identify $\mathrm {Lie} (\mathcal {U}_{\mathcal {MT}(\mathbb {Z})})$ with its image, and abusively call it the motivic Lie algebra.

The Lie algebra $\mathfrak {g}^{\mathfrak {m}}$ is non-canonically isomorphic to the free Lie algebra with one generator $\sigma _{2i+1}$ in each degree $-(2i+1)$ for $i \geq 1$.

3.1 The Ihara coaction

For any graded Hopf algebra $H$, let $IH= H_{>0}/H^2_{>0}$ denote the Lie coalgebra of indecomposable elements of $H$, and let $\pi : H_{>0} \rightarrow IH$ denote the natural map. Dualizing (2.8) (and making it into a left coaction) gives an infinitesimal coaction

(3.2)\begin{equation} \mathcal{O}({}_0 \Pi_{1}) \longrightarrow I \mathcal{O}({}_0 \Pi_{1}) \otimes_{\mathbb{Q}} \mathcal{O}({}_0 \Pi_{1}) . \end{equation}

Let $D_{r}: \mathcal {O}({}_0 \Pi _{1}) \rightarrow I \mathcal {O}({}_0 \Pi _{1})_r \otimes _{\mathbb {Q}} \mathcal {O}({}_0 \Pi _{1})$ denote its component of degree $(r,\cdot )$, and let us denote the element $e^{a_1}\ldots e^{a_n}$ in $\mathcal {O}({}_0 \Pi _{1})$ by $\mathbb {I}(0;a_1,\ldots , a_n;1)$, where $a_i \in \{0,1\}$.

Proposition 3.3 Set $a_0=0, a_{n+1}=1$. For all $r\geq 1$, and $a_1,\ldots , a_n \in \{0,1\}$,

(3.3)\begin{align} D_r \, \mathbb{I}( 0; a_1, \ldots, a_n; 1 ) =\sum_{p=0}^{n-r} \pi( \mathbb{I}(a_{p} ;a_{p+1},.\, .\,, a_{p+r}; a_{p+r+1})) \otimes \mathbb{I}(0; a_1, .\, .\,, a_{p}, a_{p+r+1}, .\, .\, , a_n ; 1) , \end{align}

where $\mathbb {I}(a_{p} ;a_{p+1},.\, .\,, a_{p+r}; a_{p+r+1}) \in \mathcal {O}({}_0 \Pi _{1})$ is defined to be zero if $a_p=a_{p+r+1}$, and equal to $(-1)^r \mathbb {I}( a_{p+r+1} ;a_{p+r},.\, .\,, a_{p+1}; a_{p})$ if $a_p=1$ and $a_{p+r+1}=0$.

Since the motivic coaction on $\mathcal {H}$ factors through the Ihara coaction, it follows that the degree $(r,\cdot )$ component factors through operators

\[ D_r: \mathcal{H} \longrightarrow I \mathcal{A} \otimes_{\mathbb{Q}} \mathcal{H} \]

given by the same formula as (3.3) in which each term $\mathbb {I}$ is replaced by its image $I^{\mathfrak {m}}$ in $\mathcal {H}$ (respectively, $\mathcal {A}$). Since $\mathcal {A}^{\mathcal {MT}}$ is cogenerated in odd degrees only, the motivic coaction on $\mathcal {H}$ is completely determined by the set of operators $D_{2r+1}$ for all $r\geq 1$ (see [Reference BrownBro12]).

4.1 Definition

The depth filtration was defined in [Reference Deligne and GoncharovDG05, § 6.1]. We recall the definition in a slightly different language. The inclusion $\mathbb {P}^1\backslash \{0,1,\infty \} \hookrightarrow \mathbb {P}^1\backslash \{0,\infty \}$ induces a map on the motivic fundamental groupoids

(4.1)\begin{equation} \pi^{\mathrm{mot}}_1(X, \overset{\rightarrow}{1}\!_0, -\overset{\rightarrow}{1}\!_1) \rightarrow \pi^{\mathrm{mot}}_1(\mathbb{G}_m, \overset{\rightarrow}{1}\!_0, 1) , \end{equation}

and hence on the de Rham realizations

\[ \mathcal{O}(\pi^{\mathrm{dR}}_1(\mathbb{G}_m, \overset{\rightarrow}{1}\!_0, 1))\longrightarrow \mathcal{O}({}_0 \Pi_{1}) , \]

and similarly for $\mathcal {O}(\pi ^{\mathrm {dR}}_1(\mathbb {G}_m, \overset {\rightarrow }{1}\!_0))\rightarrow \mathcal {O}({}_0 \Pi _{0})$. They are given by the inclusion of $\mathbb {Q}\langle e^0 \rangle$ into $\mathbb {Q} \langle e^0, e^1 \rangle$. Define their images to be $\mathfrak {D}_0 \mathcal {O}({}_0 \Pi _{1})$ and $\mathfrak {D}_0 \mathcal {O}({}_0 \Pi _{0})$, respectively. Now define increasing filtrations $\mathfrak {D}_d$ on $\mathcal {O}({}_0 \Pi _{0})$ and $\mathcal {O}({}_0 \Pi _{1})$ by induction on $d \geq 0$ as follows: $\mathfrak {D}_d$ is the largest subspace such that $\Delta \mathfrak {D}_d \subset \sum _{i+j=d} \mathfrak {D}_i\otimes \mathfrak {D}_j$, where $\Delta ^{\mathrm {dec}}: \mathcal {O}({}_0 \Pi _{x}) \rightarrow \mathcal {O}({}_0\Pi _{x}) \otimes \mathcal {O}({}_0 \Pi _{0})$ is dual to composition of paths, for $x=0,1$. Since $\Delta ^{\mathrm {dec}}$, on identifying $\mathcal {O}({}_0 \Pi _{0}) \cong \mathcal {O}({}_0 \Pi _{1}) \cong \mathbb {Q}\langle e_0,e_1\rangle$, is nothing other than the deconcatenation coproduct, the filtration $\mathfrak {D}_d \mathcal {O}({}_0 \Pi _{1})$ is given by

(4.2)\begin{equation} \mathfrak{D}_d \mathcal{O}({}_0 \Pi_{1}) = \langle \hbox{words } w \hbox{ such that } \deg_{e^1} w \leq d \rangle_{\mathbb{Q}} , \end{equation}

with respect to which $\mathcal {O}({}_0 \Pi _{1})$ is a filtered comodule over the filtered Hopf algebra $\mathcal {O}({}_0 \Pi _{0})$, with respect to $\Delta ^{\mathrm {dec}}$. Furthermore, since the map (4.1) is motivic, it follows that $\mathfrak {D}_0$ is preserved by the action of the motivic Galois group. The same is true for all $d$ by induction: if $g \in G^{\mathrm {dR}}_{\mathcal {MT}(\mathbb {Z})}$ and $g\, \mathfrak {D}_i \subset \mathfrak {D}_i$ for all $i < d$, then we also have $g \mathfrak {D}_d \subset \mathfrak {D}_d$ by definition of $\mathfrak {D}_d$, since $g$ commutes with $\Delta ^{\mathrm {dec}}$, which is motivic. Therefore the depth is also motivic and descends to the algebra $\mathcal {H}$. By Definition 3.1 the depth filtration $\mathfrak {D}_d \mathcal {H}$ is the increasing filtration defined by

\[ \mathfrak{D}_d \mathcal{H} = \langle \zeta^{{\mathfrak{m}}}(n_1,\ldots, n_i): i\leq d\rangle_{\mathbb{Q}} . \]

Following [Reference DeligneDel10], it is convenient to define the $\mathfrak {D}$-degree on $\mathcal {O} ({}_0 \Pi _{1})$ to be the degree in $e^1$. It defines a grading on ${}_0 \Pi _{1}$ which is not motivic. By (4.2), the depth filtration (which is motivic) is simply the increasing filtration associated to the $\mathfrak {D}$-degree.

4.2 Depth-graded motivic multiple zeta values

Definition 4.1 Let $\mathfrak {D}_d \mathcal {A}$ be the induced filtration on the quotient $\mathcal {A} = \mathcal {H}/\zeta ^{{\mathfrak {m}}}(2) \mathcal {H}$. We define the depth-graded motivic multiple zeta value

\[ \zeta^{{\mathfrak{m}}}_{\mathfrak{D}} (n_1,\ldots, n_r) \]

to be the image of $\zeta ^{{\mathfrak {m}}}(n_1,\ldots , n_r)$ in $\mathrm {gr}_r^{\mathfrak {D}} \mathcal {H}$.

Proposition 4.2 There is a non-canonical isomorphism of bigraded vector spaces:

(4.3)\begin{equation} \mathrm{gr}^{\mathfrak{D}} \mathcal{H} \cong \mathrm{gr}^{\mathfrak{D}} \mathcal{A} \otimes_{\mathbb{Q}} \mathbb{Q}[\zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(2)] , \end{equation}

where $\zeta ^{{\mathfrak {m}}}_{\mathfrak {D}}(2)^n$ are in depth $1$ for all $n\geq 1$.

Proof. Choose a homomorphism $\pi _2: \mathcal {H} \rightarrow \mathbb {Q}[\zeta ^{{\mathfrak {m}}}(2)]$ which respects the weight-grading and such that $\pi _2(\zeta ^{{\mathfrak {m}}}(2))=\zeta ^{{\mathfrak {m}}}(2)$. Such a homomorphism exists by [Reference Deligne and GoncharovDG05, § 5.20] (see [Reference BrownBro12, § 2.3]). Composing with the coaction (3.1), we obtain a map

\[ \mathcal{H} \overset{\Delta^{\mathcal{M}}}{\longrightarrow} \mathcal{A} \otimes_{\mathbb{Q}}\mathcal{H} \overset{id \otimes \pi_2}{\longrightarrow} \mathcal{A} \otimes_{\mathbb{Q}}\mathbb{Q}[\zeta^{{\mathfrak{m}}}(2)] . \]

By a motivic version [Reference BrownBro12] of Euler's theorem, $\zeta ^{{\mathfrak {m}}}(2)^n$ is a rational multiple of $\zeta ^{{\mathfrak {m}}}(2n)$, and so all powers of $\zeta ^{{\mathfrak {m}}}(2)$ have depth 1. The depth filtration thus satisfies

\[ \mathbb{Q}=\mathfrak{D}_0\mathbb{Q}[\zeta^{{\mathfrak{m}}}(2)] \subset \mathfrak{D}_1 \mathbb{Q}[\zeta^{{\mathfrak{m}}}(2)] = \mathbb{Q}[\zeta^{{\mathfrak{m}}}(2)] . \]

Since $\mathfrak {D}_0 \mathcal {H}=\mathbb {Q}$, it follows trivially that $\pi _2$ respects the depth filtration because it is $\mathbb {Q}$-linear. Since the depth filtration is motivic, it is also respected by $\Delta ^{\mathcal {M}}$, and therefore the map $(id \otimes \pi _2) \,\Delta ^{\mathcal {M}}: \mathcal {H}\rightarrow \mathcal {A} \otimes _{\mathbb {Q}}\mathbb {Q}[\zeta ^{{\mathfrak {m}}}(2)]$ defined above also respects the depth filtration. The statement (4.3) follows since we know that this map is an isomorphism (by [Reference BrownBro12, (2.13)] or [Reference DeligneDel10, Proposition 5.8]).

Since $\mathrm {gr}^\mathfrak {D} \mathcal {A}$ is a commutative graded Hopf algebra (for the coproduct induced by $\Delta ^{\mathcal {M}}$), it is a polynomial algebra. The same is then true for $\mathrm {gr}^{\mathfrak {D}} \mathcal {H}$, by (4.3). If $I$ denotes indecomposable elements, then it follows that $I (\mathrm {gr}^{\mathfrak {D}} \mathcal {A})\cong \mathrm {gr}^{\mathfrak {D}} \, I \mathcal {A}$.

4.3 Depth-graded motivic Lie algebra

The depth filtration defines a decreasing filtration $\mathfrak {D}^r$ on $\mathfrak {g}^{\mathfrak {m}}$ where $\mathfrak {D}^r$ consists of words with at least $r$ occurrences of $e_1$:

\[ \mathfrak{D}^r \mathfrak{g}^{\mathfrak{m}} = \langle w \in \mathfrak{g}^{\mathfrak{m}} : \deg_{e_1} w \geq r\rangle . \]

We denote the associated graded Lie algebra by $\mathfrak {dg}^{\mathfrak {m}}$. There is correspondingly a decreasing depth filtration on $\mathcal {U} \mathfrak {g}^{\mathfrak {m}}$, also denoted by $\mathfrak {D}$.

It follows from the definitions above that $\mathfrak {dg}^{\mathfrak {m}}$ is the bigraded Lie algebra dual to the bigraded coalgebra $I (\mathrm {gr}^{\mathfrak {D}} \mathcal {A})\cong \mathrm {gr}^{\mathfrak {D}} \, I \mathcal {A}$ of depth-graded motivic multiple zeta values modulo products. Thus the problem of studying relations between (motivic) multiple zeta values modulo lower depth (and modulo $\zeta ^{{\mathfrak {m}}}(2)$) and the algebra $\mathfrak {dg}^{\mathfrak {m}}$ are equivalent.

4.4 Depth-parity

The following proposition is a consequence of Tsumura's result on double shuffle equations (Proposition 6.4).

Equivalently, if $n_1+\cdots + n_r \not \equiv r \pmod 2$, and $n_1+\cdots + n_r >2$ then

(4.4)\begin{equation} \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(n_1,\ldots, n_r) \equiv 0 \pmod{\hbox{products}} . \end{equation}

We do not need to work modulo $\zeta ^{{\mathfrak {m}}}_{\mathfrak {D}}(2)$ in the previous equation since the weight is larger than $2$ by assumption and all other even zeta values $\zeta ^{{\mathfrak {m}}}(2n)$, $n\geq 2$, are products.

4.5 Depth spectral sequence

The depth filtration on the motivic Lie algebra $\mathfrak {g}^{\mathfrak {m}}$ induces a homology spectral sequence which converges to the associated graded for the depth of the homology of $\mathfrak {g}^{\mathfrak {m}}$. By Theorem 1.1, the latter satisfies

\[ H_i(\mathfrak{g}^{\mathfrak{m}} ) = \begin{cases} \bigoplus\limits_{n\geq 1} \mathbb{Q} [\sigma_{2n+1}] & \hbox{if } i =1 \\ 0 & \hbox{if } i \geq 1 \end{cases} \]

and is entirely concentrated in depth $1$.

The depth spectral sequence satisfies

\[ E^1_{{-}p,q} = \mathrm{gr}_{\mathfrak{D}}^p H_{q-p} (\mathfrak{dg}^{\mathfrak{m}}), \]

where $E^1_{-p,q}$ vanishes unless $p\geq 1$ and $p< q \leq 2p$, as can easily be seen from the Chevalley–Eilenberg chain complex which computes the homology of a Lie algebra (see § 10.2 and (10.3)). The differentials satisfy

\[ d^r_{p,q} : E^r_{p,q} \rightarrow E^{r}_{p-r,q+r-1} . \]

Here is a picture of the potentially non-vanishing $E^1=E^2$ terms:

In fact, we know by Theorem 1.1 that

\[ \mathrm{gr}^1_{\mathfrak{D}}H_1 (\mathfrak{dg}^{\mathfrak{m}}) \cong H_1(\mathfrak{g}^{\mathfrak{m}} ) \cong \bigoplus_{n\geq 1} \mathbb{Q} [\sigma_{2n+1}] , \]

and therefore everything to the left of the column indexed $-1$ converges to zero. Furthermore, one knows that $\mathrm {gr}^2_{\mathfrak {D}} H_1 (\mathfrak {dg}^{\mathfrak {m}})=0$, and $\mathrm {gr}^3_{\mathfrak {D}} H_i (\mathfrak {dg}^{\mathfrak {m}})$ vanish for all $i$. One also has a complete description of $\mathrm {gr}^2_{\mathfrak {D}} H_2(\mathfrak {dg}^{\mathfrak {m}})$ in terms of period polynomials of cusp forms, as discussed below. Therefore, the first interesting part of the differential is

\[ d^2: \mathrm{gr}^2_{\mathfrak{D}} H_2(\mathfrak{dg}^{\mathfrak{m}}) \longrightarrow \mathrm{gr}^4_{\mathfrak{D}}H_1(\mathfrak{dg}^{\mathfrak{m}}) , \]

and the main Conjecture 1 can be reformulated as saying that the components of all other differentials in the depth spectral sequence vanish.

5.1 Reminders on the standard relations

We briefly review the double shuffle relations and their depth-linearized versions. See [Reference CartierCar02, Reference RacinetRac02, Reference BrownBro17b] for further background.

5.1.1 Shuffle product

Consider the algebra $\mathbb {Q}\langle e_0, e_1 \rangle$ of words in the two letters $e_0, e_1$, equipped with the shuffle product $\, \unicode{x0448} \,$ (§ 2.1). It is defined recursively by

(5.1)\begin{equation} e_i w \, \unicode{x0448} \, e_j w' = e_i (w \, \unicode{x0448} \, e_j w') + e_j (e_i w \, \unicode{x0448} \, w') \end{equation}

for all $w,w'\in \{e_0,e_1\}^\times$ and $i,j\in \{0,1\}$, and the property that the empty word $1$ satisfies $1\, \unicode{x0448} \, w=w\, \unicode{x0448} \, 1=w$. It is a Hopf algebra for the deconcatenation coproduct. A linear map $\Phi : \mathbb {Q}\langle e_0, e_1 \rangle \rightarrow \mathbb {Q}$ is a homomorphism for the shuffle multiplication, or $\Phi _{w} \Phi _{w'} = \Phi _{w \, \unicode{x0448} \, w'}$ for all $w,w'\in \{e_0,e_1\}^\times$ and $\Phi _1=1$, if and only if the series

\[ \Phi=\sum_w \Phi_w w \in \mathbb{Q} \langle \langle e_0,e_1 \rangle \rangle \]

is invertible and group-like for the (completed) coproduct $\Delta _{\!\, \unicode{x0448} \,}$ with respect to which $e_0$ and $e_1$ are primitive (compare § 2.1). In other words, there is an equivalence:

(5.2)\begin{equation} \Phi \hbox{ homomorphism for } \, \unicode{x0448} \, \quad \Longleftrightarrow \quad \Phi \in \mathbb{Q} \langle \langle e_0,e_1 \rangle \rangle^{{\times}} \hbox{ and } \Delta_{\!\, \unicode{x0448} \,} \Phi= \Phi \hat{\otimes} \Phi. \end{equation}

One says that $\Phi$ satisfies the shuffle relations if either of the equivalent conditions (5.2) holds. Passing to the corresponding Lie algebra, we have an equivalence

(5.3)\begin{equation} \Phi_{w\, \unicode{x0448} \, w'} = 0 \hbox{ for all } w,w' \quad \Longleftrightarrow \quad \Phi \in \mathbb{Q} \langle \langle e_0,e_1 \rangle \rangle \hbox{ and } \Delta_{\!\, \unicode{x0448} \,} \Phi= 1 \hat{\otimes} \Phi + \Phi \hat{\otimes} 1. \end{equation}

One says that $\Phi$ satisfies the shuffle relations modulo products if either of the equivalent conditions (5.3) holds.

The algebra $\mathbb {Q}\langle e_0, e_1\rangle$ is bigraded for the degree, or weight (for which $e_0,e_1$ both have degree $-1$), and the $\mathfrak {D}$-degree for which $e_1$ has degree $1$, and $e_0$ degree $0$. The relations (5.1)–(5.3) evidently respect both gradings. In this case, then, passing to the depth grading does not change the relations in any way, and the linearized shuffle relations are identical to the shuffle relations modulo products.

Definition 5.1 Let $\Phi \in \mathrm {gr}_{\mathfrak {D}}^r T(e_0,e_1)$ be of weight $N$. It defines a linear form $w\mapsto \Phi _w$ on words of weight $N\geq 2$ and $\mathfrak {D}$-degree $r$. It satisfies the linearized shuffle relations if

\[ \Delta'_{\!\, \unicode{x0448} \,}\Phi = 0, \]

where $\Delta '_{\!\, \unicode{x0448} \,}$ is the reduced coproduct $\Delta '_{\!\, \unicode{x0448} \,}(\Phi )= \Delta _{\!\, \unicode{x0448} \,}(\Phi ) - 1 \otimes \Phi - \Phi \otimes 1$. Equivalently, $\Phi _{w\, \unicode{x0448} \, w'} = 0$ for all words $w,w'\in \{e_0,e_1\}^{\times }$ of total weight $N$ and total $\mathfrak {D}$-degree $r$.

5.1.2 Stuffle product

Let $Y=\{ \mathsf {y} _n, n\geq 1\}$ denote an alphabet with one letter $\mathsf {y} _i$ in every degree $\geq 1$, and consider the graded algebra $\mathbb {Q}\langle Y\rangle$ equipped with the stuffle product [Reference RacinetRac02]. It is defined recursively by

(5.4)\begin{equation} \mathsf{y} _i w * \mathsf{y} _j w' = \mathsf{y} _i (w * \mathsf{y} _j w') + \mathsf{y} _j (\mathsf{y} _i w * w') + \mathsf{y} _{i+j} (w * w') \end{equation}

for all $w,w'\in Y^\times$ and $i,j \geq 1$, and the property that the empty word $1$ satisfies $1* w= w* 1=w$. A linear map $\Phi : \mathbb {Q}\langle Y \rangle \rightarrow \mathbb {Q}$ is a homomorphism for the stuffle multiplication, or $\Phi _{w} \Phi _{w'} = \Phi _{w * w'}$ for all $w,w'\in Y^{\times }$ and $\Phi _1=1$, if and only if

\[ \Phi=\sum_w \Phi_w w \in \mathbb{Q} \langle \langle Y \rangle \rangle^{{\times}} \]

is group-like for the (completed) coproduct $\Delta _{*} : \mathbb {Q} \langle \langle Y \rangle \rangle \rightarrow \mathbb {Q} \langle \langle Y \rangle \rangle \hat {\otimes }_{\mathbb {Q}} \mathbb {Q} \langle \langle Y \rangle \rangle$ which is a homomorphism for the concatenation product and defined on generators by

(5.5)\begin{equation} \Delta_{*} \mathsf{y} _n = \sum_{i=0}^n \mathsf{y} _i \otimes \mathsf{y} _{n-i} . \end{equation}

Thus the stuffle relations are equivalent to being group-like for $\Delta _{*}$:

(5.6)\begin{equation} \Phi \hbox{ homomorphism for } * \quad \Longleftrightarrow \quad \Phi \in \mathbb{Q} \langle \langle Y \rangle \rangle^{{\times}} \hbox{ and } \Delta_{*} \Phi= \Phi \hat{\otimes} \Phi. \end{equation}

One says that $\Phi$ satisfies the stuffle relations if either of the equivalent conditions (5.6) holds. Passing to the corresponding Lie algebra, we have an equivalence

(5.7)\begin{equation} \Phi_{w* w'} = 0 \hbox{ for all } w,w' \quad \Longleftrightarrow \quad \Phi \in \mathbb{Q} \langle \langle Y\rangle \rangle \hbox{ and } \Delta_{*} \Phi= 1 \hat{\otimes} \Phi + \Phi \hat{\otimes} 1. \end{equation}

One says that $\Phi$ satisfies the stuffle relations modulo products if either of the equivalent conditions (5.7) holds.

The algebra $\mathbb {Q}\langle Y\rangle$ is graded for the degree (where $\mathsf {y} _n$ has degree $n$), and filtered for the depth (where $\mathsf {y} _n$ has depth $1$). By inspection of (5.4), we notice that the rightmost term is of lower depth than the other terms and therefore drops out in the associated graded. The associated graded of $*$ therefore satisfies the same recursive definition as for $\, \unicode{x0448} \,$ and it follows that the associated depth-graded

(5.8)\begin{equation} \mathrm{gr}_{\mathfrak{D}} ( \mathbb{Q} \langle Y \rangle, *) \cong (\mathbb{Q} \langle Y \rangle, \, \unicode{x0448} \,) \end{equation}

is simply the shuffle algebra on $Y$. The depth induces a decreasing filtration on the (dual) completed Hopf algebra $\mathbb {Q}\langle \langle Y \rangle \rangle$, and it follows from (5.5) that the images of the elements $\mathsf {y} _n$ are primitive in the associated graded. Thus

(5.9)\begin{equation} \mathrm{gr}_{\mathfrak{D}} \Delta_{*} = \Delta_{\, \unicode{x0448} \,,Y}, \end{equation}

where $\Delta _{\, \unicode{x0448} \,,Y}: \mathbb {Q}\langle \langle Y \rangle \rangle \rightarrow \mathbb {Q}\langle \langle Y \rangle \rangle \hat {\otimes }_{\mathbb {Q}} \mathbb {Q}\langle \langle Y \rangle \rangle$ is the (completed) coproduct for which the elements $\mathsf {y} _n$ are primitive, and which is a homomorphism for concatenation.

Definition 5.2 Let $\Phi \in T(Y)$, the tensor (co)algebra on $Y$, of degree $N\geq 2$ and $\mathfrak {D}$-degree $r$. It defines a linear map $w\mapsto \Phi _w$ on words in $Y$ of weight $N$ and $\mathfrak {D}$-degree $r$. We say that it satisfies the linearized stuffle relations if

\[ \Delta'_{\, \unicode{x0448} \,, Y} \Phi = 0, \]

where $\Delta '_{\, \unicode{x0448} \,, Y}= \Delta _{\, \unicode{x0448} \,,Y} - 1 \otimes \mathrm {id} - \mathrm {id} \otimes 1$ is the reduced coproduct of $\Delta _{\, \unicode{x0448} \,, Y}$ for which the $\mathsf {y} _n$ are primitive. Equivalently, $\Phi _{w\, \unicode{x0448} \, w'} = 0$ for all words $w,w'\in Y$ of total weight $N$ and total $\mathfrak {D}$-degree $r$.

5.1.3 Linearized double shuffle relations

In order to consider both relations simultaneously, define a linear map

\[ \alpha: \mathbb{Q} \langle e_0, e_1 \rangle\rightarrow \mathbb{Q}\langle Y \rangle \]

which maps every word beginning in $e_0$ to $0$, and such that

\[ \alpha (e_1 e_0^{a_1}\ldots e_1 e_0^{a_r} ) = \mathsf{y} _{a_1+1}\ldots \mathsf{y} _{a_r+1} . \]

In [Reference RacinetRac02], Racinet considered a certain graded vector space, denoted $\mathfrak {dm}_0(\mathbb {Q})$ [Reference RacinetRac02, Définition 2.4], of series satisfying the shuffle and stuffle relations and a regularization condition, and showed that it is a Lie algebra for the Ihara bracket. Since we consider the depth-graded version of this algebra, the regularization plays no role here.

Definition 5.3 Let $\Phi \in \mathrm {gr}^{\mathfrak {D}}_r T(e_0,e_1)$ of weight $N$. The linear form $w\mapsto \Phi _w$ on words of weight $N$ and $\mathfrak {D}$-degree $r$ satisfies the linearized double shuffle relations if

\[ \Delta_{\!\, \unicode{x0448} \,}' \Phi = 0 \quad \hbox{and}\quad \Delta'_{\, \unicode{x0448} \,,Y} \, \alpha(\Phi) =0 . \]

When $r=1$, we add the extra condition that $\Phi =0$ when $N$ is even, and $\Phi (e_0)=\Phi (e_1)=0$. Let $\mathfrak {ls} \subset \mathcal {U} \mathfrak {g}$ denote the vector space of elements satisfying the linearized double shuffle relations. It is bigraded by weight and depth.

Remark 5.4 There is a natural inclusion

(5.10)\begin{equation} \mathrm{gr}_{\mathfrak{D}} \mathfrak{dm}_0(\mathbb{Q}) \longrightarrow \mathfrak{ls} . \end{equation}

The graded subspace of $\mathfrak {ls}$ of weight $N$ and depth $d$ is isomorphic to the vector space denoted $D_{N+d,d}$ in [Reference Ihara, Kaneko and ZagierIKZ06]. In [Reference Ihara, Kaneko and ZagierIKZ06] it is conjectured that (5.10) is an isomorphism.

Racinet proved that $\mathfrak {dm}_0(\mathbb {Q})$ is preserved by the Ihara bracket. Since the latter is homogeneous for the $\mathfrak {D}$-degree, it follows that $\mathrm {gr}_{\mathfrak {D}} \mathfrak {dm}_0(\mathbb {Q})$ is also preserved by the bracket, but this is not quite enough to prove that $\mathfrak {ls}$ is too.

Theorem 5.5 The bigraded vector space $\mathfrak {ls}$ is preserved by the Ihara bracket.

Proof. The compatibility of the shuffle product with the Ihara bracket follows by definition. It therefore suffices to check that the linearized stuffle relation is preserved by the bracket. The proof of [Reference RacinetRac02] goes through identically, and uses in an essential way the fact that the images of the elements $\mathsf {y} _{2n}$ in $\mathrm {gr}^1_{\mathfrak {D}} \mathfrak {dm}_0(\mathbb {Q})$ are zero (which in $\mathfrak {dm}_0(\mathbb {Q})$ follows from [Reference RacinetRac02, Proposition 2.2], but holds in $\mathfrak {ls}$ by Definition 5.3).

Many thanks to a referee for pointing out at least two places in the literature which have subsequently provided a more detailed proof of this statement: [Reference MaassaraniMaa19] and also [Reference SchnepsSch15, Theorem 3.4.3]. It would be interesting to know if a suitable linearized version of the associator relations is equivalent to the linearized double shuffle relations.

5.2 Summary of definitions

We have the following Lie subalgebras of the Lie algebra $\mathfrak {g}\cong (\mathbb {L}(e_0,e_1), \{\,{,}\,\})$, equipped with the Ihara bracket:

\[ \mathfrak{g}^{\mathfrak{m}} \subseteq \mathfrak{dm}_0(\mathbb{Q}) , \]

where $\mathfrak {g}^{\mathfrak {m}}$ is the image of the (weight-graded) Lie algebra of $\mathcal {U}_{\mathcal {MT}(\mathbb {Z})}$ and is isomorphic to the free graded Lie algebra on (non-canonical) generators $\sigma _{2i+1}$ for $i\geq 1$. A standard conjecture states that $\mathfrak {g}^{\mathfrak {m}} \subseteq \mathfrak {dm}_0(\mathbb {Q})$ is an isomorphism. Passing to the depth-graded versions, and writing $\mathfrak {dg}^{\mathfrak {m}}= \mathrm {gr}_{\mathfrak {D}}\mathfrak {g}^{\mathfrak {m}}$, we have inclusions of bigraded Lie algebras

(5.11)\begin{equation} \mathfrak{dg}^{\mathfrak{m}}\subseteq \mathrm{gr}_{\mathfrak{D}} \, \mathfrak{dm}_0(\mathbb{Q}) \subseteq \mathfrak{ls} , \end{equation}

where $\mathfrak {ls}$ stands for the linearized double shuffle algebra. Once again, all Lie algebras in (5.11) are conjectured to be equal. The bigraded dual space $(\mathfrak {dg}^{\mathfrak {m}})^{\vee }$ is isomorphic to the Lie coalgebra of depth-graded motivic multiple zeta values, modulo $\zeta ^{{\mathfrak {m}}}(2)$ and modulo products. The Lie algebras (5.11) are Lie subalgebras of $\mathrm {gr}_{\mathfrak {D}} \mathfrak {g} \cong \mathfrak {g}$, since the Ihara bracket is homogeneous with respect to $\mathfrak {D}$-degree.

All the above Lie algebras can, in particular, be viewed inside the vector space $\mathcal {U} \mathbb {L}(e_0,e_1)= T(\mathbb {Q} e_0 \oplus \mathbb {Q} e_1)$, which is graded for the $\mathfrak {D}$-degree. Next, we show that complicated expressions in the non-commutative algebra $T(\mathbb {Q} e_0 \oplus \mathbb {Q} e_1)$ can be greatly simplified by encoding words of fixed $\mathfrak {D}$-degree in terms of polynomials.

6.1 Composition of polynomials

Recall from (2.1) that $\mathcal {U} \,\mathbb {L}(e_0,e_1)$ is isomorphic to the bigraded tensor algebra $T(e_0,e_1)$, and the space $\mathrm {gr}_{\mathfrak {D}}^r \, \mathcal {U}\, \mathbb {L}(e_0,e_1)=\mathrm {gr}_{\mathfrak {D}}^r \, T(e_0,e_1)$ of $\mathfrak {D}$-degree $r$ is the spanned by words in $e_0,e_1$ with exactly $r$ occurrences of $e_1$.

Definition 6.1 Consider the isomorphism of vector spaces

(6.1)\begin{equation} \begin{gathered} \rho: \mathrm{gr}_{\mathfrak{D}}^r \, T(e_0,e_1) \longrightarrow \mathbb{Q}[y_0,\ldots, y_r] \\ e_0^{a_0} e_1 e_0^{a_1} e_1 \ldots e_1 e_{0}^{a_{r}} \mapsto y_0^{a_0}y_1^{a_1}\ldots y_r^{a_r}. \end{gathered} \end{equation}

It maps elements of degree $n$ to elements of degree $n-r$.

The operator $\, \underline {\circ }\, : T(e_0,e_1) \otimes _{\mathbb {Q}} T(e_0,e_1) \rightarrow T(e_0,e_1)$ respects the $\mathfrak {D}$-grading, so defines a map

(6.2)\begin{equation} \begin{aligned} \, \underline{\circ}\, : \mathbb{Q}[y_0,\ldots, y_r] \otimes_{\mathbb{Q}} \mathbb{Q}[y_0,\ldots, y_s] & \longrightarrow \mathbb{Q}[y_0,\ldots, y_{r+s}] \\ f(y_0,\ldots, y_r) \otimes g(y_0,\ldots, y_s) & \mapsto f\, \underline{\circ}\, g\, ( y_0,\ldots, y_{r+s}) \end{aligned} \end{equation}

which can be read off from (2.7). Explicitly, it is

(6.3)\begin{align} f \, \underline{\circ}\, g (y_0,\ldots, y_{r+s}) &= \sum_{i=0}^s f(y_i,y_{i+1}, \ldots, y_{i+r}) g(y_0,\ldots, y_i, y_{i+r+1}, \ldots, y_{r+s}) \nonumber\\ &\quad + ({-}1)^{\deg f + r} \sum_{i=1}^s f(y_{i+r},\ldots, y_{i+1},y_i) g(y_0,\ldots, y_{i-1}, y_{i+r}, \ldots, y_{r+s}). \end{align}

Antisymmetrizing, and using Proposition 2.2, we obtain a formula for the Ihara bracket

(6.4)\begin{equation} \begin{gathered} \mathfrak{g} \wedge \mathfrak{g} \longrightarrow \mathfrak{g} \\ \{ f, g\} = f \, \underline{\circ}\, g - g \, \underline{\circ}\, f . \end{gathered} \end{equation}

The linearized double shuffle relations on words translate into functional equations for polynomials after applying the map $\rho$. We describe some of these below.

6.2 Translation invariance

The additive group $\mathbb {G}_a$ acts on $\mathbb {A}^{r+1}$ by translation, and hence $\mathbb {G}_a(\mathbb {Q})$ acts on its ring of functions via $\lambda : (y_0,\ldots , y_r) \mapsto (y_0+\lambda ,\ldots , y_r+\lambda )$.

Proof. The subspace $\mathfrak {g}^{\mathfrak {m}} \subset \mathfrak {g} \cong \mathbb {L}(e_0,e_1) \subset T(e_0,e_1)$ is contained in the subspace of elements which are primitive with respect to the shuffle coproduct $\Delta _{\, \unicode{x0448} \,}$. Let $\pi _0: T(e_0,e_1) \rightarrow \mathbb {Q}$ denote the linear map which sends the word $e_0$ to $1$ and all other words to $0$, and consider the map $\partial _0 = (\pi _0 \otimes id) \circ \Delta _{\, \unicode{x0448} \,}$. It defines a derivation $\partial _0: T(e_0,e_1) \longrightarrow T(e_0,e_1)$ which satisfies

\[ \partial_0 (e_0^{a_0} e_1 \ldots e_1 e_0^{a_r} )= \sum_{i=0}^r a_i \, e_0^{a_0} e_1 \ldots e_1 e_0^{a_i-1}e_1 \ldots e_1 e_0^{a_r} \]

for all non-negative integers $a_0,\ldots , a_r$. Let $r\geq 1$, $\xi \in \mathrm {gr}_{\mathfrak {D}}^r T(e_0,e_1)$, and $f=\rho (\xi )$. If $\xi$ is primitive for $\Delta _{\, \unicode{x0448} \,}$, it satisfies $\partial _0 \xi =0$ and the previous equation is

(6.5)\begin{equation} \sum_{i=0}^r \frac{ \partial f }{\partial y_i }=0 . \end{equation}

This is equivalent to $({\partial }/{\partial \lambda }) g=0$, where $g= f(y_0,\ldots , y_r) - f(y_0+\lambda ,\ldots , y_r+\lambda )$. Therefore $g$ is constant in $\lambda$. It vanishes at $\lambda =0$, so $g\equiv 0$ and $f$ is translation invariant.

Let us denote the map which sends $y_0$ to zero and $y_i$ to $x_i$ for $i=1,\ldots , r$ by

(6.6)\begin{equation} \begin{gathered} \mathbb{Q}[y_0,\ldots, y_r] \longrightarrow \mathbb{Q}[x_1,\ldots, x_r] \\ f \mapsto \bar{f}, \end{gathered} \end{equation}

where $\bar {f}$ is the ‘reduced’ polynomial $\bar {f}(x_1,\ldots , x_r)= f(0,x_1,\ldots , x_r)$. In the case when $f$ is translation invariant, we can retrieve $f$ from $\bar {f}$ via

(6.7)\begin{equation} f(y_0,\ldots, y_r)=\bar{f}(y_1-y_0,\ldots, y_r-y_0). \end{equation}

Taking the coefficients of (6.7) gives equation I2 of [Reference BrownBro12], which gives a formula for the shuffle-regularization of iterated integrals which begin with any sequence of $0$'s.

To avoid confusion, we reserve the variables $x_1,\ldots , x_r$ for the reduced polynomial $\bar {\rho }(\xi )$ and use the variables $y_0, \ldots , y_r$ as above to denote the full polynomial $\rho (\xi )$.

6.3 Antipodal symmetries

The set of primitive elements in a Hopf algebra is stable under the antipode. For the shuffle Hopf algebra this is the map $* : T(e_0,e_1) \rightarrow T(e_0,e_1)$ which sends $w \mapsto (-1)^{|w|} \tilde {w}$, where $\tilde {w}$ is the reversed word and $|w|$ the length of $w$. Restricting to $\mathfrak {D}$-degree $r$ and transporting via $\rho$, we obtain a map

(6.8)\begin{equation} \begin{gathered} \sigma: \mathbb{Q}[y_0,\ldots, y_r] \longrightarrow \mathbb{Q}[y_0,\ldots, y_r] \\ \sigma( f) (y_0,\ldots, y_r) = ({-}1)^{\deg(f) +r} f(y_r,\ldots, y_0). \end{gathered} \end{equation}

Therefore, if $f\in \mathbb {Q}[y_0,\ldots , y_r]$ satisfies the shuffle relations (5.3), then

(6.9)\begin{equation} f+\sigma(f)=0 . \end{equation}

Since the stuffle algebra, graded for the depth filtration, is isomorphic to the shuffle algebra on $Y$ (5.9), it follows that its antipode is the map $\mathsf {y} _{i_1}\ldots \mathsf {y} _{i_r} \mapsto (-1)^r\mathsf {y} _{i_r}\ldots \mathsf {y} _{i_1}$. This defines an involution

(6.10)\begin{equation} \begin{gathered} \bar{\tau}: \mathbb{Q}[x_1,\ldots, x_r] \longrightarrow \mathbb{Q}[x_1,\ldots, x_r] \\ \bar{\tau}(\bar{f}) (x_1,\ldots, x_r) = ({-}1)^r \bar{f}(x_r,\ldots, x_1). \end{gathered} \end{equation}

Therefore if $\bar {f}\in \mathbb {Q}[x_1,\ldots , x_r]$ satisfies the linearized stuffle relations, then $\bar {f} + \bar {\tau }(\bar {f})=0$. Note that the involution $\bar {\tau }$ lifts to an involution

(6.11)\begin{equation} \begin{gathered} \tau: \mathbb{Q}[y_0,y_1,\ldots, y_r] \longrightarrow \mathbb{Q}[y_0, y_1,\ldots, y_r] \\ \tau(f) (y_0,y_1,\ldots, y_r) = ({-}1)^r f(y_0,y_r,\ldots, y_1), \end{gathered} \end{equation}

and therefore if $f\in \mathbb {Q}[y_0,\ldots , y_r]$ satisfies both translational invariance and the linearized stuffle relations, we have

(6.12)\begin{equation} f+ \tau(f)=0 . \end{equation}

The composition $\tau \sigma$ is the signed cyclic rotation of order $r+1$,

\[ \tau \sigma(f) (y_0,\ldots, y_r) = ({-}1)^{\deg f} f(y_r,y_0,\ldots, y_{r-1}), \]

and plays an important role in the rest of this paper.

6.4 Parity relations

The following result is well known, and was first proved by Tsumura [Reference TsumuraTsu04], and subsequently in ([Reference Ihara, Kaneko and ZagierIKZ06, Theorem 7]). We repeat the proof for convenience.

Proof. Let $f\in \mathbb {Q}[y_0,\ldots , y_r]$ be in the image of $\rho (\mathfrak {dg}^{\mathfrak {m}}_r)$. In particular, it satisfies the linearized stuffle relations (6.9) and (6.12). Following [Reference Ihara, Kaneko and ZagierIKZ06], consider the relation

\[ \mathsf{y} _1 \, \unicode{x0448} \, \mathsf{y} _2 \ldots \mathsf{y} _r = \mathsf{y} _1\ldots \mathsf{y} _r + \sum_{i=2}^{r} \mathsf{y} _2 \ldots \mathsf{y} _i \mathsf{y} _1 \mathsf{y} _{i+1} \ldots \mathsf{y} _r , \]

where $r\geq 2$ (the case $r=1$ follows from Definition 5.3). Then we have

\[ f(y_0,y_1,\ldots, y_r) + \sum_{i=2}^{r} f(y_0,y_2,\ldots, y_i, y_1, y_{i+1}, \ldots, y_r) =0 . \]

Now apply the automorphism of $\mathbb {Q}[y_0,\ldots , y_r]$ defined on generators by $y_i \mapsto y_{i+1}$, where $i$ is taken modulo $r+1$. This leads to the equation

\[ f(y_1,y_2,\ldots, y_r, y_0) + \sum_{i=3}^{r+1} f(y_1,y_3,\ldots, y_i, y_2, y_{i+1}, \ldots,y_r, y_0) =0 . \]

By applying a cyclic rotation $\tau \sigma$ to each term, we get

\[ f(y_0, y_1,y_2,\ldots, y_r) + \sum_{i=3}^{r} f(y_0, y_1,y_3,\ldots, y_i, y_2, y_{i+1}, \ldots, y_r) + f(y_2, y_1, y_3,\ldots, y_r,y_0) =0 . \]

The first two terms can be interpreted as the terms occurring in the linearized stuffle product ($\mathsf {y} _2 \, \unicode{x0448} \, \mathsf {y} _1\mathsf {y} _3 \ldots \mathsf {y} _r$) minus the first term. As a result, one obtains the equation

\[{-}f(y_0, y_2,y_1, y_3,\ldots, y_r)+f(y_2, y_1, y_3,\ldots, y_r,y_0) =0 , \]

which, by a final application of $\tau \sigma$ to the right-hand term, yields

\[ (({-}1)^{\deg f} -1 )f(y_0, y_2,y_1, y_3,\ldots, y_r) =0 . \]

Therefore, in the case when $\deg f$ is odd, the polynomial $f$ must vanish.

6.5 Dihedral symmetry and the Ihara bracket

For all $r\geq 1$, consider the graded vector space $\mathfrak {p} _r$ of polynomials $f\in \mathbb {Q}[y_0,\ldots , y_r]$ which satisfy

(6.13)\begin{equation} \begin{gathered} f(y_0,\ldots, y_r) = f({-}y_0,\ldots,- y_r), \\ f + \sigma(f) = f+ \tau (f) = 0 . \end{gathered} \end{equation}

The maps $\sigma ,\tau$ generate a dihedral group $D_{r+1}=\langle \sigma , \tau \rangle$ of symmetries acting on $\mathfrak {p} _r$ of order $2r+2$, and any element $f$ satisfying (6.13) is invariant under cyclic rotation:

\[ f = \sigma \tau (f),\quad \hbox{where}\ (\sigma \tau)^{r+1} = id , \]

and anti-invariant under the reflections $\sigma$ and $\tau$. Thus $\mathbb {Q} f \subset \mathfrak {p} _r$ is isomorphic to the one-dimensional sign (orientation) representation $\varepsilon$ of the dihedral group $D_{r+1}$.

Proposition 6.5 Suppose that $f\in \mathfrak {p} _r$ and $g\in \mathfrak {p} _s$ are polynomials satisfying (6.13). Then the Ihara bracket is the signed average over the dihedral symmetry group:

(6.14)\begin{equation} \{f,g\} = \sum_{\mu \in D_{r+s+1}} \varepsilon(\mu) \mu\big( f(y_0,y_1,\ldots, y_r) g(y_{r},y_{r+1}\ldots, y_{r+s})\big). \end{equation}

In particular, $\{.\,{,}\,.\}: \mathfrak {p} _r \times \mathfrak {p} _s \rightarrow \mathfrak {p} _{r+s}$, and $\mathfrak {p} =\bigoplus _{r\geq 1} \mathfrak {p} _r$ is a bigraded Lie algebra.

Proof. A straightforward calculation from (6.3) and definition (6.4) gives

\[ \{f,g\} = \sum_{i } f(y_i, y_{i+1}, \ldots, y_{i+r}) \big( g(y_{i+r}, y_{i+r+1}, \ldots,y_{i-1})- g(y_{i+r+1}, y_{i+r+2}, \ldots,y_i) \big), \]

where the summation indices are taken modulo $r+s+1$. Invoking dihedral symmetry of $f, g$ leads to formula (6.14). The Jacobi identity for $\{.\,{,}\,.\}$ is automatic since the Ihara action is an action, but can also be proved very easily by identifying its terms with the set of double cuts in a polygon (see [Reference BrownBro17b]). The fact that parity (first equation of (6.13)) is preserved by $\{\,{,}\,\}$ is clear. The anti-invariance under $\sigma , \tau$ is also clear from the dihedral symmetry of (6.14).

A similar dihedral symmetry was also found by Goncharov [Reference GoncharovGon01b]; the interpretation of the dihedral reflections as antipodes may or may not be new. Since the Ihara action is, by definition, compatible with the shuffle product, it follows from Lemma 6.2 that translation invariance is preserved by the Ihara bracket. One can also easily verify this by direct computation:

\[ \sum_{i=0}^r \frac{\partial f }{\partial y_i }=\sum_{i=0}^s \frac{\partial g }{\partial y_i }= 0 \quad \Longrightarrow \quad \sum_{i=0}^{r+s} \frac{\partial \{f,g\} }{\partial y_i } =0 . \]

By abuse of notation, we can equivalently view $\bar {\mathfrak {p} }_r$ as the subspace of polynomials $\bar {f}\in \mathbb {Q}[x_1,\ldots , x_r]$ whose lift $\bar {f}(y_1 -y_0,\ldots , y_r-y_0)$ lies in $\mathfrak {p} _r$. Explicitly, $\bar {\mathfrak {p} }_r$ is the vector space of polynomials satisfying

1. (1) $\bar {f}(x_1,\ldots , x_r )= \bar {f}(-x_1,\ldots , -x_r )$,

2. (2) $\bar {f}(x_1,\ldots , x_r ) + (-1)^r \bar {f}(x_r,\ldots , x_1 ) =0$,

3. (3) $\bar {f}(x_1,\ldots , x_r ) + (-1)^r \bar {f}(x_{r-1}-x_r,\ldots , x_1-x_r, -x_r ) =0$,

with Lie bracket induced by (6.14). To conclude the previous discussion, the map

\[ \bar{\rho} : \mathfrak{dg}^{\mathfrak{m}} \longrightarrow \bar{\mathfrak{p} } \]

is an injective map of bigraded Lie algebras.

6.6 Generators in depth $1$ and examples

It follows from Theorem 1.1 that in depth $1$, the Lie algebra $\mathfrak {dg}^{\mathfrak {m}}$ has exactly one generator in every odd weight $\geq 3$:

\[ \bar{\rho} (\mathfrak{dg}^{\mathfrak{m}}_1 ) = \bigoplus_{n\geq 1} \mathbb{Q} \, x_1^{2n} . \]

In particular, the algebras $\mathfrak {dg}^{\mathfrak {m}} \subset \mathrm {gr}_{\mathfrak {D}} \mathfrak {dm}_0 (\mathbb {Q}) \subset \mathfrak {ls}$ are all isomorphic in depth $1$.

Definition 6.8 Denote the Lie subalgebra generated by $x_1^{2n}$, for $n\geq 1$, by

(6.15)\begin{equation} \mathfrak{dg}^{\mathrm{odd}} \subset \mathfrak{dg}^{\mathfrak{m}} . \end{equation}

Example 6.9 The formula for the Ihara bracket in depth 2 can be written

\[ \{ x_1^{2m}, x_1^{2n}\} = x_1^{2m} \big(x_2^{2n}- (x_2-x_1)^{2n} \big) + (x_2-x_1)^{2m} \big( x_1^{2n} - x_2^{2n}\big)+ x_2^{2m} \big( (x_2-x_1)^{2n}- x_1^{2n} \big). \]

6.7 Relations between depth-graded motivic multiple zeta values

We briefly explain how we may describe all relations between $\zeta ^{{\mathfrak {m}}}_{\mathfrak {D}}$ using this formalism.

Proposition 4.2 states that $\mathrm {gr}^{\mathfrak {D}} \mathcal {H}$ is the free bigraded right $\mathcal {U} \mathfrak {dg}^{\mathfrak {m}}$-module generated by the $\zeta ^{{\mathfrak {m}}}_{\mathfrak {D}}(2n)$, and can be represented by polynomials as follows. The role of the even zeta value $\zeta ^{{\mathfrak {m}}}(2n)$, for $n\geq 1$, is played by the depth-graded associator element $\tau ^{(1)}$ in weight $2n$ constructed in [Reference BrownBro17a, §§ 7.3 and 7.4], and is encoded by the monomial in one variable:

\[ \bar{\tau}_{2n} = x_1^{2n-1}. \]

Therefore, a relation

\[ \sum_{n_1+ \cdots+ n_d=N} \lambda_{n_1,\ldots, n_d}\, \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(n_1,\ldots, n_d) = 0 \]

holds between (shuffle-regularized) depth-graded motivic multiple zeta values in depth $d$ and weight $N$ if and only if $\sum _{\underline {n} = (n_1,\ldots , n_d)} \lambda _{\underline {n}} c_{\underline {n}} =0$ for all

\[ \xi = \sum_{n_1+ \cdots+ n_d=N} c_{n_1,\ldots, n_d}\, x_1^{n_1-1}\ldots x_{d}^{n_d-1} , \]

where $\xi$ is of depth $d$ and weight $N$ of the form

\begin{align*} x&= g_1 \, \underline{\circ}\, \big( g_2 \, \underline{\circ}\, \big( \ldots \ldots \big( g_{d-1} \, \underline{\circ}\, g_d \big)\! \cdots \!\big)\big) \quad \hbox{ if } N\equiv d \pmod{2},\\ x&= g_1 \, \underline{\circ}\, \big( g_2 \, \underline{\circ}\, \big( \ldots \ldots \big( g_{d-1} \, \underline{\circ}\, \bar{\tau}_{2k} \big)\! \cdots \!\big)\big)\quad \hbox{ if } N\not\equiv d \pmod{2} , \end{align*}

in which the $g_i$ are the polynomial representatives of generators of $\mathfrak {dg}^{\mathfrak {m}}$ and $k\geq 1$.

Examples 6.10 We give some simple examples in depth 2. For this, our formula for $\, \underline {\circ }\,$ gives, for any $m,n\geq 1$,

(6.16)\begin{equation} x_1^{2m} \, \underline{\circ}\, x_1^{n} = x_1^{2m} x^{n}_2 + (x_2-x_1)^{2m} \big( x^{n}_1 - x^{n}_2\big) . \end{equation}

(1) In weight 5, depth 2, $\mathrm {gr}_2^{\mathfrak {D}} \mathcal {H}_5$ is one-dimensional generated by the element $\bar {\sigma }_3 \circ \bar {\tau }_2$, which we can compute using (6.16) to be

\[ x_1^2 \, \underline{\circ}\, x_1 = x_1^3 - 2x_1^2x_2 +3 x_1 x_2^2 - x_2^3 . \]

It encodes the coefficients of $\zeta ^{{\mathfrak {m}}}(3) \zeta ^{{\mathfrak {m}}}(2)$ in the following (shuffle-regularized) depth-graded multiple zeta values in depth 2 and weight 4, as one can see from the relations

\begin{align*} \zeta^{{\mathfrak{m}}}(3,2) &= \tfrac{9}{2}\, \zeta^{{\mathfrak{m}}}(5) - 2\, \zeta^{{\mathfrak{m}}}(3) \zeta^{{\mathfrak{m}}}(2), \\ \zeta^{{\mathfrak{m}}}(2,3) &={-}\tfrac{11}{2}\, \zeta^{{\mathfrak{m}}}(5) +3\, \zeta^{{\mathfrak{m}}}(3) \zeta^{{\mathfrak{m}}}(2), \\ \zeta^{{\mathfrak{m}}}(1,4) &= 2 \, \zeta(5) - \zeta^{{\mathfrak{m}}}(3) \zeta^{{\mathfrak{m}}}2). \end{align*}

For instance, we may read off the relation $3\, \zeta ^{{\mathfrak {m}}}_{\mathfrak {D}}(3,2)+2\, \zeta ^{{\mathfrak {m}}}_{\mathfrak {D}}(2,3)=0$ from $x_1^2 \, \underline {\circ }\, x_1$.

(2) In weight 7, depth 2, $\mathrm {gr}_2^{\mathfrak {D}} \mathcal {H}_7$ is two-dimensional generated by the two elements $\bar {\sigma }_5 \circ \bar {\tau }_2$ and $\bar {\sigma }_3 \circ \bar {\tau }_4$, which we compute via

\begin{align*} x_1^4 \, \underline{\circ}\, x_1 &= x_1^5 - 4\,x_1^4 x_2 + 10 \, x_1^3 x_2^2 - 10 \, x_1^2 x_2^3 +5 \, x_1x_2^4 - x_2^5 \\ x_1^2 \, \underline{\circ}\, x^3_1 &= x_1^5 - 2 \, x_1^4 x_2 + x_1^3 x_2^2 +2 \, x_1 x_2^4 - x_2^5 . \end{align*}

The coefficients in these expressions encode the coefficients of $\zeta ^{{\mathfrak {m}}}(5) \zeta ^{{\mathfrak {m}}}(2)$ and $\zeta ^{{\mathfrak {m}}}(3) \zeta ^{{\mathfrak {m}}}(4)$ in depth 2 multiple zeta values of weight 7. For example,

\begin{align*} \zeta^{{\mathfrak{m}}}(5,2) &= 10\, \zeta^{{\mathfrak{m}}}(7) - 4\, \zeta^{{\mathfrak{m}}}(5) \zeta^{{\mathfrak{m}}}(2) - 2 \, \zeta^{{\mathfrak{m}}}(3) \zeta^{{\mathfrak{m}}}(4), \\ \zeta^{{\mathfrak{m}}}(4,3) &={-}18\, \zeta^{{\mathfrak{m}}}(7) +10 \, \zeta^{{\mathfrak{m}}}(5) \zeta^{{\mathfrak{m}}}(2) + \zeta^{{\mathfrak{m}}}(3) \zeta^{{\mathfrak{m}}}(4), \\ \zeta^{{\mathfrak{m}}}(3,4) &= 17 \, \zeta^{{\mathfrak{m}}}(7) - 10 \, \zeta^{{\mathfrak{m}}}(5) \zeta^{{\mathfrak{m}}}(2), \\ \zeta^{{\mathfrak{m}}}(2,5) &={-}11 \, \zeta^{{\mathfrak{m}}}(7) +5 \, \zeta^{{\mathfrak{m}}}(5) \zeta^{{\mathfrak{m}}}(2) +2 \,\zeta^{{\mathfrak{m}}}(3) \zeta^{{\mathfrak{m}}}(4) . \end{align*}

Choosing any linear functional on the coefficients of $x_1^4 x_2$, $x_1^3 x_2^2$ and $x_1^2 x_2^3$ which vanishes on $x_1^4 \, \underline {\circ }\, x_1$ and $x_1^2 \, \underline {\circ }\, x^3_1$ leads to a relation between depth-graded motivic multiple zeta values, for instance:

\[ \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(5,2) + 2 \, \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(4,3) + \tfrac{8}{5}\, \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(3,4) = 0 . \]

(3) In weight 8, depth 2, $\mathrm {gr}_2^{\mathfrak {D}} \mathcal {H}_8$ is two-dimensional generated by the pair of elements $\bar {\sigma }_3 \circ \bar {\sigma }_5$ and $\bar {\sigma }_5 \circ \bar {\sigma }_3$, which we compute using (6.16) by

\begin{align*} x_1^4 \, \underline{\circ}\, x^2_1 &= x_1^6 - 2 \, x_1^5 x_2 + x_1^4 x_2^2 +2 \, x_1 x_2^5 - x_2^6 \\ x_1^2 \, \underline{\circ}\, x^4_1 &= x_1^6 - 4 \, x_1^5 x_2 + 6 \, x_1^4 x_2^2 - 5\, x_1^2 x_2^4 + 4\, x_1 x_2^5 - x_2^6. \end{align*}

From these, we deduce all relations between depth-graded motivic multiple zeta values of weight 8 and depth 2, such as

\[ 2\, \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(3,5) +5 \, \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(2,6) + 10\, \zeta^{{\mathfrak{m}}}_{\mathfrak{D}}(1,7) = 0 \]

since $2.( 0) + 5.(2) + 10. (-1) =0$ and $2.(-5) + 5.(4) + 10.(-1)=0$ where the numbers in brackets are the coefficients of $x_1^2x_2^4$, $x_1 x_2^5$, $x_2^6$ in $x_1^4 \, \underline {\circ }\, x^2_1$ and $x_1^2 \, \underline {\circ }\, x^2_3$. Indeed, one may check that $5 \, \zeta ^{{\mathfrak {m}}}(2,6) + 2\, \zeta ^{{\mathfrak {m}}}(3,5) + 10\, \zeta ^{{\mathfrak {m}}}(1,7) = \frac {6}{175} \zeta ^{{\mathfrak {m}}}(2)^4$.

7.1 Reminders on period polynomials

We recall some definitions from [Reference Kohnen and ZagierKZ84, § 1.1]. Let $S_{2k}(\mathrm {SL}_2(\mathbb {Z}))$ denote the space of cusp forms of weight $2k$ for the full modular group $\mathrm {SL}_2(\mathbb {Z})$.

Definition 7.1 Let $n\geq 1$ and let $W_{2n}^e\subset \mathbb {Q}[X,Y]$ denote the vector space of homogeneous polynomials $P(X,Y)$ of degree $2n-2$ satisfying

(7.1)\begin{gather} P(X,Y) + P(Y,X)= 0 , \quad P({\pm} X, \pm Y) = P (X, Y), \end{gather}

(7.2)\begin{gather}P(X, Y) + P(X-Y,X) + P ({-}Y, X-Y) =0 . \end{gather}

The space $W_{2n}^e$ contains the polynomial $p_{2n}= X^{2n-2}-Y^{2n-2}$, and is a direct sum

\[ W_{2n}^e \cong \mathsf{S} _{2n} \oplus \mathbb{Q} \,p_{2n}, \]

where $\mathsf {S} _{2n}$ is the subspace of polynomials which vanish at $(X,Y)=(1,0)$. We write $\mathsf {S} = \bigoplus _{n} \mathsf {S} _{2n}$. The Eichler–Shimura–Manin theorem gives a map which associates, in particular, an even-period polynomial to every cusp form:

\[ S_{2k}(\mathrm{SL} _2(\mathbb{Z})) \longrightarrow W_{2k}^e\otimes_{\mathbb{Q}}\mathbb{C} . \]

Explicitly, if $f$ is a cusp form of weight $2k$, the map is given by

(7.3)\begin{equation} f \quad \mapsto \quad \bigg( \int_{0}^{i \infty} f(z) (X-z Y)^{2k-2}\, dz\bigg)^{+}, \end{equation}

where $+$ denotes the projection onto invariants under the involution $(X,Y)\mapsto (X,-Y)$, that is, $a^+(X,Y)= \frac {1}{2} (a (X,Y) + a(X,-Y))$. The three-term equation (7.2) follows from integrating around the geodesic triangle with vertices $0,1,i \infty$ and is reminiscent of the hexagon equation for associators. The map (7.3) is an isomorphism onto a subspace of $W^e_{2k} \otimes \mathbb {C}$ of codimension $1$. Thus $\dim S_{2k}(\mathrm {SL} _2(\mathbb {Z})) = \dim \mathsf {S} _{2k}$ and it follows from classical results on the space of modular forms of level 1 that

(7.4)\begin{equation} \sum_{n\geq 0} \dim \mathsf{S} _{2n} s^{2n} = \frac{s^{12} }{(1-s^4)(1-s^6)} = \mathbb{S}(s) . \end{equation}

7.2 Linearized double shuffle in depths 2 and 3

We first recall from [Reference Ihara, Kaneko and ZagierIKZ06] that $\mathbb {D}_2$ is the graded space of homogeneous polynomials $f\in \mathbb {Q}[x_1,x_2]$ in two variables satisfying the linearized double shuffle equations in depth 2:

\[ f(x_1,x_2) + f(x_2,x_1) =0 \quad \hbox{and} \quad f(x_1,x_1+x_2) + f(x_2, x_1+x_2) =0 . \]

The space $\mathbb {D}_3$ consists of homogeneous polynomials $f \in \mathbb {Q}[x_1,x_2,x_3]$ such that

\begin{align*} f(x_1,x_2,x_3) + f(x_2,x_1,x_3) +f(x_2,x_3,x_1) & = 0, \\ f^{\sharp}(x_1,x_2,x_3) + f^{\sharp}(x_2,x_1,x_3) +f^{\sharp}(x_2,x_3,x_1) & = 0 , \end{align*}

where $f^{\sharp }(x,y,z) = f(x,x+y, x+y+z)$. The general double shuffle equations and their linearized versions are derived in [Reference BrownBro17b, §§ 4–7] using Hopf-algebra-theoretic techniques.

7.3 A short exact sequence in depth 2

The Ihara bracket gives a map

(7.5)\begin{equation} \{ \,{,}\,\} : \mathfrak{ls} _1 \wedge \mathfrak{ls} _1 \longrightarrow \mathfrak{ls} _2 . \end{equation}

Applying the isomorphism $\bar {\rho }$ leads to a map

(7.6)\begin{equation} \mathbb{D}_1 \wedge \mathbb{D}_1 \longrightarrow \mathbb{D}_2 , \end{equation}

given by the formula in Example 6.9. Since $\mathbb {D}_1$ is isomorphic to the graded vector space $\mathbb {Q}[x_1^{2n},{n\geq 1}]$, it follows that $\mathbb {D}_1 \wedge \mathbb {D}_1$ is isomorphic to the space of antisymmetric even polynomials $p(x_1,x_2)$ with positive bidegrees, with basis $x_1^{2m} x_2^{2n} - x_1^{2n} x_2^{2m}$ for $m>n>0$. It follows from Example 6.9 that the image of $p(x_1,x_2)$ under (7.6) is

\[ p(x_1,x_2) + p(x_2-x_1, x_1)+p({-}x_2, x_1-x_2). \]

Comparing with (7.2) and (7.1), we immediately deduce (cf. [Reference Ihara and TakaoIT93, Reference GoncharovGon05, Reference Gangl, Kaneko and ZagierGKZ06, Reference SchnepsSch06]) that

(7.7)\begin{equation} \ker ( \{ \,{,}\,\} : \mathbb{D}_1 \wedge \mathbb{D}_1 \longrightarrow \mathbb{D}_2 ) \overset{\sim}{\longrightarrow} \mathsf{S} . \end{equation}

In fact, the dimensions of the space $\mathbb {D}_2$ have been computed several times in the literature (e.g. by some simple representation-theoretic arguments), and it is relatively easy to show [Reference Gangl, Kaneko and ZagierGKZ06] that the following sequence is exact:

(7.8)\begin{equation} 0 \longrightarrow \mathsf{S} \longrightarrow \mathbb{D}_1 \wedge \mathbb{D}_1 \longrightarrow \mathbb{D}_2 \longrightarrow 0 . \end{equation}

Example 7.2 The smallest non-trivial period polynomial occurs in degree 10 and is given by $s_{12} = X^2Y^2(X-Y)^3(X+Y)^3 = X^8 Y^2 -3 X^6 Y^4+3 X^4 Y^6-X^2Y^8.$ By the exact sequence (7.8) it immediately gives rise to the equation

(7.9)\begin{equation} 3 \{x_1^4,x_1^6\} = \{x_1^2,x_1^8\} , \end{equation}

which, by the faithfulness of the map $\bar {\rho }$, is equivalent to Ihara's formula (1.7).

7.4 A short exact sequence in depth 3

If $V$ is a vector space let $\mathrm {Lie}_n(V) \subset V^{\otimes n}$ denote the component of degree $n$ in the free graded Lie algebra $\mathrm {Lie}(V)$ generated by $V$, viewed inside the tensor algebra $T(V)= \bigoplus _{n\geq 0} V^{\otimes n}$. It is the subspace spanned by Lie brackets of the form $[v_1,[v_2,\ldots , [v_{n-1},v_n]\ldots ]$ where the Lie bracket on $T(V)$ is the antisymmetrization of the tensor multiplication (i.e.$[v_1,v_2]=v_1v_2-v_2v_1$). Thus, as a graded vector space, $\mathrm {Lie}(V) \cong \bigoplus _{n \geq 1} \mathrm {Lie}_n(V)$, and each $\mathrm {Lie}_n$ is a direct sum of Schur functors (e.g. $\mathrm {Lie}_1(V) = V$ and $\mathrm {Lie}_2(V) = {\bigwedge} ^2 V$).

The triple Ihara bracket gives a trilinear map

\[ \mathrm{Lie}_3 (\mathfrak{ls} _1) \longrightarrow \mathfrak{ls} _3 , \]

and hence a map $\mathrm {Lie}_3(\mathbb {D}_1) \rightarrow \mathbb {D}_3$ whose image is spanned by $\{x_1^{2a} , \{ x_1^{2b}, x_1^{2c}\} \!\}$, for $a,b,c\geq 1$. Goncharov has studied the space $\mathbb {D}_3$, and computed its dimensions in each weight [Reference GoncharovGon05]. It follows from his work that the sequence

(7.10)\begin{equation} 0 \longrightarrow \mathsf{S} \otimes_{\mathbb{Q}} \mathbb{D}_1 \longrightarrow \mathrm{Lie}_3 (\mathbb{D}_1) \longrightarrow \mathbb{D}_3\longrightarrow 0 \end{equation}

is exact, where the first map (identifying $\mathsf {S}$ with $\ker (\Lambda ^2 \mathbb {D}_1 \rightarrow \mathbb {D}_2)$) is given by

\[ \mathsf{S} \otimes_{\mathbb{Q}} \mathbb{D}_1 \hookrightarrow \Lambda^2 \mathbb{D}_1 \otimes_{\mathbb{Q}} \mathbb{D}_1 \rightarrow \mathrm{Lie}_3(\mathbb{D}_1) , \]

and the second map in the sequence immediately above is $[a,b]\otimes c \mapsto [c, [a,b]]$. Starting from depth $4$, the structure of $\mathfrak {ls} _d\cong \mathbb {D}_d$ is not known.Footnote ⁶ In particular, it is easy to show that the map given by the quadruple Ihara bracket

\[ \mathrm{Lie}_4 (\mathbb{D}_1) \longrightarrow \mathbb{D}_4 \]

is not surjective, since in weight $12$, $\dim \mathbb {D}_4= 1$, but $\mathrm {Lie}_4 (\mathbb {D}_1) = 0$. Our next purpose is to construct the missing elements in depth 4.

8.1 Linearized equations in depth 4

For the convenience of the reader, we write out the linearized double shuffle relations in full in depth 4. There are four equations. In order to write them down we shall use the following notation, where $f \in \mathbb {Q}[x_1,\ldots , x_4]$, and we are given any set of indices $\{i,j,k,l\} = \{1,2,3,4\}$:

(8.1)\begin{align} \begin{gathered} f(ijkl) = f(x_i,x_j,x_k,x_l), \\ f^{\sharp}(ijkl) = f(x_i,x_i+x_j,x_i+x_j+x_k,x_i+x_j+x_k+x_l) . \end{gathered} \end{align}

Then $\mathbb {D}_4$ (see [Reference Ihara, Kaneko and ZagierIKZ06, § 8]) is the subspace of polynomials $f\in \mathbb {Q}[x_1,\ldots , x_4]$ satisfying

(8.2)\begin{gather} f(1\, \unicode{x0448} \, 234)=0 , \quad f(12\, \unicode{x0448} \, 34)=0, \end{gather}

(8.3)\begin{gather}f^{\sharp}(1\, \unicode{x0448} \, 234)=0 , \quad f^{\sharp}(12\, \unicode{x0448} \, 34)=0 , \end{gather}

where $f$ and $f^{\sharp }$ are extended by linearity in the obvious way, and

\begin{gather*} 1 \, \unicode{x0448} \, 234 = 1234 + 2134 + 2314+ 2341 ,\\ 12 \, \unicode{x0448} \, 34 = 1234 + 1324 + 1342+ 3124 + 3142 + 3412 . \end{gather*}

For example, the first equation of (8.2) is simply

\[ f(x_1,x_2,x_3,x_4) + f(x_2,x_1,x_3,x_4) + f(x_2,x_3,x_1,x_4) + f(x_2,x_3,x_4,x_1)=0 . \]

We construct some exceptional solutions to these equations from period polynomials.

8.2 Definition of the exceptional elements

Let $f(x,y) \in \mathsf {S} _{2n+2}$ be an even-period polynomial of degree $2n$ which vanishes at $y=0$. It follows from (7.1) and (7.2) that it vanishes along $x=0$ and $x-y=0$. Therefore we can write

\[ f(x,y)=xy (x-y) f_0(x,y), \]

where $f_0(x,y)\in \mathbb {Q}[x,y]$ is symmetric and homogeneous of degree $2n-3$, and satisfies

(8.4)\begin{equation} f_0(x,y) + f_0(y-x,-x) + f_0({-}y,x-y) = 0 . \end{equation}

Let us also write $f_1(x,y)= (x-y) f_0(x,y)$. We have $f_1(-x,y)=f_1(x,-y)=-f_1(x,y)$.

Definition 8.1 Let $f\in \mathbb {Q}[x,y]$ be an even-period polynomial as above. Define

(8.5)\begin{align} \begin{gathered} \mathsf{e}_f \in \mathbb{Q}[y_0,y_1,y_2,y_3,y_4], \\ \mathsf{e}_f = \sum_{\mathbb{Z}/ 5 \mathbb{Z} } \big( f_1(y_4-y_3,y_2-y_1) + (y_0-y_1) f_0 (y_2-y_3,y_4-y_3) \big) , \end{gathered} \end{align}

where the sum is over cyclic permutations $(y_0,y_1,y_2,y_3, y_4 ) \mapsto (y_1,y_2,y_3, y_4, y_0)$. Its reduction $\bar {\mathsf {e}}_f\in \mathbb {Q}[x_1,\ldots , x_4]$ is obtained by setting $y_0=0, y_i =x_i$, for $i=1,\ldots , 4$.

Remark 8.2 The full expression for $\bar {e}_f$ is explicitly

(8.6)\begin{align} \bar{e}_f(x_1,x_2,x_3,x_4) &= f_1(x_4-x_3,x_2-x_1) + f_1({-}x_4,x_3-x_2) + f_1(x_1,x_4-x_3) \nonumber\\ &\quad + f_1(x_2-x_1,-x_4) +f_1(x_3-x_2,x_1) -x_1 f_0 (x_2-x_3,x_4-x_3) \nonumber\\ &\quad+ (x_1-x_2) f_0 (x_3-x_4,-x_4) + (x_2-x_3) f_0 (x_4, x_1) \nonumber\\ &\quad + (x_3-x_4) f_0 ({-}x_1, x_2-x_1)+ x_4 f_0 (x_1-x_2, x_3-x_2) . \end{align}

Since $f$ is even it vanishes to order $2$ along $x=0,y=0,x=y$. Therefore

\[ f_0(0,y) = f_0(x,0) = f_0(x,x)=0, \]

and the same holds a fortiori for $f_1$. If we set $x_3=x_4=0$ in equation (8.6) then all terms except for the fifth vanish and we are left with

(8.7)\begin{equation} \bar{\mathsf{e}}_f( x,y, 0,0) = f_1 ({-}y,x)= f_1(x,y) . \end{equation}

In this way, the period polynomial $f$ can be retrieved from $\bar {\mathsf {e}}_f$: it is $xy\bar {\mathsf {e}}_f(x,y,0,0)$. This computation is related to the discussion in [Reference Blümlein, Broadhurst and VermaserenBBV10, § 9.2], regarding multiple zeta values of depth 4 of the form $\zeta (1,1,m, n)$.

8.3 Exceptional elements and linearized double shuffle equations

Theorem 8.3 The reduced polynomial $\bar {\mathsf {e}}_f$ obtained from (8.5) satisfies the linearized double shuffle relations. In particular, we have an injective linear map

\[ \bar{\mathsf{e}}:\mathsf{S} \longrightarrow \mathbb{D}_4 . \]

Proof. The injectivity follows immediately from (8.7). The proof that the linearized double shuffle relations hold is a finite computation. In the absence of a purely conceptual proof, we shall break the calculation into more easily verifiable pieces.

We first consider the stuffle equations. It follows from general properties of the Dynkin operator on shuffle algebras that a homogeneous polynomial in four variables satisfies the two linearized stuffle equations (8.2) if and only if it is in the image of the map $\lambda :\mathbb {Q}[x_1,\ldots , x_4] \rightarrow \mathbb {Q}[x_1,\ldots , x_4]$, with eight terms defined by

\[ \lambda (f ) (x_1,\ldots, x_4) = \alpha (f)(x_1,x_2, x_3, x_4) -\alpha (f)(x_4,x_3, x_2, x_1), \]

where $\alpha ( f)(x_1,\ldots , x_4)$ is the linear combination

\[ f(x_1,x_2,x_3,x_4) - f(x_1,x_2,x_4,x_3) - f(x_1,x_4, x_2, x_3) + f(x_1,x_4,x_3,x_2) . \]

For a detailed discussion and proofs, we refer to [Reference BrownBro17b, § 16.4 and Corollary 16.6]. The linearized stuffle relations (8.2) hold for the sum of the first five terms in $f_1$, and for the sum of the second five terms in $f_0$ in (8.6) separately. Consider first the terms in $f_1$. They consist of two parts:

\[ T_1= f_1(x_4-x_3,x_2-x_1) \]

and

\[ T_2= f_1({-}x_4,x_3-x_2) + f_1(x_1,x_4-x_3) + f_1(x_2-x_1,-x_4) +f_1(x_3-x_2,x_1) . \]

One easily checks that $\lambda (T_1) = 4 \, T_1$, and that $\lambda (f_1(x_1,x_2-x_3))$ equals

\begin{align*} &f_1(x_1,x_2-x_3) - f_1(x_1,x_2-x_4) - f_1(x_1,x_4-x_2) + f_1(x_1,x_4-x_3) \\ &\qquad - f_1(x_4,x_3-x_2) + f_1(x_4,x_3-x_1) + f_1(x_4,x_1-x_3) - f_1(x_4,x_1-x_2) = 4\, T_2 , \end{align*}

using only the fact that $f_1$ is antisymmetric and odd in $x$ and $y$. Thus $T_1, T_2$ lie in the image of $\lambda$ and are solutions to the linearized stuffle equations.

Now consider the terms in $f_0$ in (8.6). Once again, they break into two parts:

\[ T_3= x_4 f_0 (x_1-x_2, x_3-x_2) -x_1 f_0 (x_2-x_3,x_4-x_3) \]

and

\[ T_4= (x_1-x_2) f_0 (x_3-x_4,-x_4) + (x_2-x_3) f_0 (x_4, x_1) + (x_3-x_4) f_0 ({-}x_1, x_2-x_1) . \]

One checks that $\lambda (T_k ) = 4 \, T_k$ for $k=3,4$ using the three-term relation (8.4), and hence $T_3,T_4$ are solutions to the linearized stuffle equations. This is the only point in the proof where the three-term relation is needed.

For the linearized shuffle relations, note that there exists $g \in \mathbb {Q}[x,y]$ such that $f_0(x,y) = (x+y) g(x,y)$ and $f_1 = (x^2-y^2) g(x,y)$ since an even-period polynomial $f(x,y)$ vanishes along $x=y$ and is even in both $x$ and $y$. The polynomial $g(x,y)$ is symmetric and odd in $x$ and in $y$ (i.e. $g(x,y)=g(y,x)$ and $g(-x,y)=g(x,-y)=-g(x,y)$). These properties suffice to prove that (8.6) satisfies the shuffle equations. The full expression again splits into two pieces with four and six terms, respectively:

\begin{align*} T_5&= (x_4^2-x_{23}^2) g(x_4, x_{23})+x_{23}(x_1+x_4)g(x_1, x_4) + x_{12}(x_{34}-x_4)g(x_{34}, -x_4)\\ &\quad +(x_{12}^2-x_4^2)g(x_{12}, x_4), \end{align*}

where we use the notation $x_{ij} = x_i-x_j$, and

\begin{align*} T_6&= x_{34}(x_{21}-x_1)g(x_1, x_{12})+x_4(x_{32}+x_{12})g(x_{12}, x_{32})+(x_{32}^2-x_1^2)g(x_{32}, x_1) \\ &\quad +(x_1^2-x_{43}^2)g(x_1, x_{43}) + x_1(x_{34}+x_{32})g(x_{32}, x_{34})+(x_{43}^2-x_{21}^2)g(x_{43}, x_{21}). \end{align*}

Both $T_5$ and $T_6$ can laboriously be verified to satisfy the two shuffle equations in depth 4. The statement follows since $\mathsf {e}_f$ equals $T_1+T_2+T_3+T_4 = T_5+T_6$.

Identifying $\mathfrak {ls} _4$ with $\mathbb {D}_4$ via the map $\bar {\rho }$, we can view $\mathsf {e}$ as a map from $\mathsf {S}$ to $\mathfrak {ls} _4$. Note that relation (7.2) is proved for the periods of modular forms by integrating round contours very similar to those which prove the symmetry and hexagonal relations for associators. It would be very interesting to see if there is any connection between the five-fold symmetry of the element $\mathsf {e}_f$ and the pentagon equation.

Example 8.4 It follows from (7.4) that the space of period polynomials in degrees $12, 16, 18$ and $20$ is of dimension $1$. Choose integral generators:

\begin{align*} f_{12} &= [x_{{1}}^{8} , x_{{2}}^{2}]-3\,[x_{{1}}^{6},x_{{2}}^{4}], \\ f_{16} &= 2\,[x_{{1}}^{12},x_{{2}}^{2}]-7\,[x_{{1}}^{10},x_{{2}}^{4}]+11\,[x_{ {1}}^{8},x_{{2}}^{6}], \\ f_{18} &= 8\, [x_{{1}}^{14},x_{{2}}^{2}]-25\,[x_{{1}}^{12},x_{{2}}^{4}]+26\,[x_ {{1}}^{10},x_{{2}}^{6}], \\ f_{20} &= 3\,[x_{{1}}^{16},x_{{2}}^{2}]-10\,[x_{{1}}^{14},x_{{2}}^{4}]+14\,[x_ {{1}}^{12},x_{{2}}^{6}]-13\,[x_{{1}}^{10},x_{{2}}^{8}], \end{align*}

where $[x_1^a,x_2^b]$ denotes $x_1^ax_2^b-x_1^bx_2^a$. Let $\mathsf {e}_{12},\ldots , \mathsf {e}_{20}$ denote the corresponding exceptional elements. We know by Theorem 1.1 that $\mathfrak {g}^{\mathfrak {m}}$ is of dimension $2$ in weight $12$, spanned by $\{ \sigma _3, \sigma _9\}$ and $\{\sigma _5, \sigma _7\}$. We know by (7.9) that in weight 12, $\mathfrak {dg}^{\mathfrak {m}}_2$ is of dimension 1, $\mathfrak {dg}^{\mathfrak {m}}_3$ vanishes by parity, so it follows that $\mathfrak {dg}^{\mathfrak {m}}_4$ is of dimension 1 and hence spanned by $\bar {\mathsf {e}}_{12}$ (since we know that $\mathfrak {ls} _4$ in weight 12 is one-dimensional). Writing out just a few of its coefficients as an example, we have:

\[ \bar{\mathsf{e}}_{12}= x_3^7 x_4 -116\, x_1^3 x_2^2x_3^2x_4 -57\, x_1^2 x_2^5x_4+\cdots \quad (118 \hbox{ terms in total}). \]

Using $\bar {\mathsf {e}}_{12}$, one can write all depth-graded motivic multiple zeta values of depth 4 and weight 12 as multiples of $\zeta _{\mathfrak {D}}(1,1,8,2)$. For example, one has

\[ \zeta_{\mathfrak{D}}(4,3,3,2 ) \equiv{-}116\, \zeta_{\mathfrak{D}}(1,1,8,2) , \quad \zeta_{\mathfrak{D}}(3,6,1,2 ) \equiv{-}57\, \zeta_{\mathfrak{D}}(1,1,8,2) \]

modulo products and modulo multiple zeta values of depth $\leq 2$.

8.4 Are the exceptional elements motivic?

We say that an exceptional element $\mathsf {e}_f$ is motivic if it lies in the depth-graded motivic Lie algebra:

\[ \mathsf{e}_f \in \mathfrak{dg}^{\mathfrak{m}}_4 \subseteq \mathfrak{ls} _4 . \]

Since $\mathfrak {dg}^{\mathfrak {m}}_k= \mathfrak {ls} _k$ for $k =1,2,3$, to prove that an exceptional element $\mathsf {e}_f$ is motivic it is enough to show that, modulo commutators, it lies in the image of the map

\[ d: \mathsf{S} \longrightarrow (\mathfrak{dg}^{\mathfrak{m}}_4)^{\mathrm{ab}}, \]

where $\mathsf {S} \subset \Lambda ^2 \mathfrak {dg}^{\mathfrak {m}}_1$ is the space of relations in depth $2$, and $d$ is the first non-trivial differential in the spectral sequence on $\mathfrak {g}^{\mathfrak {m}}$ associated to the depth filtration (§ 4.5).

The map $d$ can be computed explicitly as follows. Choose a lift $\sigma _{2n+1} \in \mathfrak {g}^{\mathfrak {m}}$ of every generator $\bar {\sigma }_{2n+1} \in \mathfrak {dg}^{\mathfrak {m}}_1$, and decompose it according to the $\mathfrak {D}$-degree:

\[ \sigma_{2n+1} = \sigma^{(1)}_{2n+1} + \sigma^{(2)}_{2n+1} + \sigma^{(3)}_{2n+1}+ \cdots , \]

where $\sigma ^{(i)}_{2n+1}$ is of $\mathfrak {D}$-degree $i$, and $\sigma ^{(1)}_{2n+1} =\bar {\sigma }_{2n+1}$. Then, for any element

\[ \xi = \sum_{i < j} \lambda_{i,j}\, \sigma_i \wedge \sigma_j \in \mathsf{S} = \ker ( \{.\,{,}\,.\}: \Lambda^2 \mathfrak{dg}^{\mathfrak{m}}_1\rightarrow \mathfrak{dg}^{\mathfrak{m}}_2) \]

with $\lambda _{i,j} \in \mathbb {Q}$, we have

\[ d \xi = \sum_{i < j} \lambda_{i,j} ( \{\sigma^{(1)}_i, \sigma^{(3)}_j\} +\{\sigma^{(2)}_i, \sigma^{(2)}_j\} + \{\sigma^{(3)}_i, \sigma^{(1)}_j\} ) , \]

where $\{.\,{,}\,. \}: {\bigwedge} ^2 \mathfrak {g}^{\mathfrak {m}} \rightarrow \mathfrak {g}^{\mathfrak {m}}$ can be computed on the level of polynomial representations by exactly the same formula as the one given in § 6.

Examples 8.5 The elements $\sigma _3, \sigma _5, \sigma _7, \sigma _9$ defined by the coefficients of $\zeta (3),\zeta (5),\zeta (7)$, and $\zeta (9)$ in weights $3$, $5$, $7$, $9$ in Drinfeld's associator are canonical, and we have

(8.8)\begin{equation} \{ \sigma_3, \sigma_9 \} -3 \{ \sigma_5, \sigma_7 \} \equiv \tfrac{691}{144}\, \mathsf{e}_{12} \mod \hbox{depth} \geq 5 , \end{equation}

which proves that the element $\mathsf {e}_{12}$ is motivic. Using the depth-parity proposition (Proposition 6.4), one can show that the corresponding congruence

\[ \{ \sigma_3, \sigma_9 \} -3 \{ \sigma_5, \sigma_7 \} \equiv 0 \mod 691, \]

propagates to depth 5 also. Compare with the ‘key example’ of [Reference IharaIha02, p. 258] and the ensuing discussion. Thereafter, one checks that

\begin{align*} d \big(2 \, \sigma_3\wedge \sigma_{13} -7 \, \sigma_5 \wedge \sigma_{11} + 11 \, \sigma_7\wedge \sigma_{9} \big) &\equiv \tfrac{3617 } {720} \mathsf{e}_{16} \pmod{ \mathfrak{a}}, \\ d \big(8 \, \sigma_3 \wedge \sigma_{15} -25 \, \sigma_5 \wedge \sigma_{13} + 26 \, \sigma_7 \wedge \sigma_{11} \big) &\equiv \tfrac{43867 }{9000} \mathsf{e}_{18} \pmod{ \mathfrak{a}}, \\ d \big(3 \, \sigma_3 \wedge \sigma_{17} -10 \, \sigma_5 \wedge \sigma_{13} + 14 \, \sigma_7 \wedge \sigma_{13} -13 \, f_9 \wedge f_{11} \big) &\equiv \tfrac{174611 }{35280} \mathsf{e}_{20} \pmod{ \mathfrak{a}}, \end{align*}

where $\mathfrak {a} = \{\mathfrak {g}^{\mathfrak {m}} ,\mathfrak {g}^{\mathfrak {m}} \} + \mathfrak {D}^5 \mathfrak {g}^{\mathfrak {m}}$, that is, the previous identities hold modulo commutators and modulo terms of depth 5 or more. In this manner, I have checked that the elements $\mathsf {e}_f$ are motivic for all $f$ up to weight $30$. In particular, it seems that the differential $d$ is related to our map $\mathsf {e}$ (which is defined over $\mathbb {Z}$) up to a non-trivial isomorphism of the space of period polynomials. The numerators on the right-hand side are the numerators of $\zeta (16)\pi ^{-16}, \zeta (18)\pi ^{-18}$, and $\zeta (20)\pi ^{-20}$. Unfortunately, it does not seem possible to construct canonical zeta elements $\sigma _{2n+1}$ for $n\geq 5$ in a consistent way such that the above relations hold exactly in $\mathfrak {g}^{\mathfrak {m}} _4$ (and not modulo $\mathfrak {a}$).

Using the theory of the unipotent fundamental group of the Tate elliptic curve, we showed in [Reference BrownBro17a] how to construct canonical elements $\sigma ^{(3)}_{2i+1}$ modulo depths $\geq 5$, which enables one to write down the differential $d$ explicitly. The only remaining difficulty in proving that the elements $\mathsf {e}_f$ are motivic is therefore to understand better the quotient $\mathfrak {ls} / \{ \mathfrak {ls} , \mathfrak {ls} \}$ in depth $4$. Yasuda has since shown, assuming that the $\mathsf {e}_f$ are motivic, how to relate the exceptional elements to the differential $d$ using the action of Hecke operators on the space of period polynomials.

If the elements $\mathsf {e}_f$ can be shown to be motivic, then they provide in particular an answer to the question raised by Ihara [Reference IharaIha02, end of § 4, p. 259]. The appearance of the numerators of Bernoulli numbers is related to [Reference IharaIha02, Conjecture 2], and the Ihara–Takao relations [Reference Ihara and TakaoIT93], and has been studied from the Galois-theoretic perspective by Sharifi [Reference SharifiSha02] and McCallum and Sharifi [Reference McCallum and SharifiMS03].

9.1 Unevenness

For any polynomial $f\in \mathbb {Q}[x_1,\ldots , x_r]$, let

\[ \pi^k_{x_i} f = \hbox{coefficient of }x_i^k \hbox{ in } f \]

and denote the projection onto the even part in $x_i$ by

\[ \pi^{\mathrm{ev}}_{x_i} \, f = \sum_{k\geq 0} (\pi^{2k}_i f)\, x_i^{2k} . \]

We can also write $\pi ^{\mathrm {ev}}_{x_i} f = \frac {1}{2} \big ( f(x_1,\ldots , x_i, \ldots , x_r) + f(x_1,\ldots , -x_i, \ldots , x_r) \big )$.

Lemma 9.1 The elements $\mathsf {e}_f(y_0, y_1, y_2, y_3, y_4)$ are uneven:

(9.1)\begin{equation} \pi_{y_0}^{\mathrm{ev}} \pi_{y_1}^{\mathrm{ev}} \pi_{y_2}^{\mathrm{ev}} \pi_{y_3}^{\mathrm{ev}} \pi_{y_4}^{\mathrm{ev}} (\mathsf{e}_f )=0 . \end{equation}

We shall see later in § 11.1 that the property of being uneven is motivic, that is, stable under the Ihara bracket, and is related to the totally odd zeta values. We conjecture that a solution to the linearized double shuffle equations is uneven if and only if it is in the Lie ideal of $\mathfrak {ls}$ generated by exceptional elements.

9.2 Sparsity

Lemma 9.2 The elements $\mathsf {e}_f(y_0,y_1,y_2,y_3,y_4)$ are sparse:

(9.2)\begin{equation} \frac{\partial^5 }{\partial y_0 \partial y_1 \partial y_2 \partial y_3 \partial y_4} (\mathsf{e}_f )=0 . \end{equation}

In other words, $\mathsf {e}_f$ is a linear combination of monomials which only depend on four out of five of the variables $y_0,\ldots , y_4$.

One can show that this property is also motivic, that is, forms an ideal under the Ihara bracket. Call an element $f\in \mathfrak {p}_r$ sparse if it is annihilated by

\[ \frac{\partial^{r+1} }{\partial y_0 \ldots \partial y_r} . \]

Proof. Let $f \in \mathfrak {p}_r$ and $g\in \mathfrak {p}_s$ where $f$ is sparse. The Ihara bracket $\{f,g\}$ is given by (6.14) and consists of a cyclic sum over terms of the form

\[ \big( f(y_0,\ldots, y_r) - f(y_{r+s}, y_0, \ldots, y_{r-1})\big) g(y_r,\ldots, y_{r+s}) , \]

to which we apply the product of ${\partial }/{\partial y_i}$ for $0\leq i \leq r+s$. Using the chain rule with respect to ${\partial }/{\partial y_r}$ and ${\partial }/{\partial y_{r+s}}$ and invoking the sparsity of $f$, we obtain

(9.3)\begin{equation} \bigg(\prod_{i=0}^{r-1} \frac{\partial}{\partial y_i}\bigg) \big( f(y_0,\ldots, y_r) - f(y_{r+s}, y_0, \ldots, y_{r-1})\big) \times \bigg( \prod_{i=r}^{r+s} \frac{\partial}{\partial y_i} \bigg) g(y_r,\ldots, y_{r+s}) . \end{equation}

By cyclic symmetry of $f$, we have

\[ f(y_{r+s}, y_0, \ldots, y_{r-1}) =f(y_0,\ldots, y_{r-1}, y_{r+s}) . \]

Using the sparsity of $f$ once more, we find that

\[ \bigg( \prod_{i=0}^{r-1} \frac{\partial}{\partial y_i}\bigg) f(y_0,\ldots, y_r) = \bigg( \prod_{i=0}^{r-1} \frac{\partial}{\partial y_i}\bigg) f( y_0, \ldots, y_{r-1},y_{r+s}) \]

since the left-hand side is a polynomial which is annihilated by ${\partial }/{\partial y_r}$, and hence does not depend on $y_r$. In particular, it equals the right-hand side. It follows that the left-hand factor of (9.3) vanishes, which completes the proof.

In fact, there are other differential equations satisfied by the $\mathsf {e}_f$ and one can use these equations to define various filtrations on the Lie algebras $\mathfrak {p}$ and $\mathfrak {ls}$. It would be interesting to try to prove that the degree in the exceptional elements defines a grading on the Lie subalgebra of $\bar {\mathfrak {p}}$ spanned by the $x_1^{2n}$ and the $\bar {\mathsf {e}}_f$, as predicted by the conjectures below.

Remark 9.4 The properties of unevenness and sparsity imply that almost all the coefficients of $\mathsf {e}_f$ are zero. This implies that the freeness of the motivic Lie algebra (Theorem 1.1) hangs by a thread (see, for example, (8.8)).

If we define the interior of a polynomial $p\in \mathbb {Q}[x_1,\ldots , x_r]$ to be $p^{o} = \pi ^{\geq 2}_1 \ldots \pi ^{\geq 2}_r p$, where $\pi _i^{\geq 2}(f) = \sum _{k\geq 2} \pi ^k_{x_i}(f) \,x_i^k$, then in fact the majority of the non-trivial monomials in $\bar {\mathsf {e}}_f$ are determined by a single term,

(9.4)\begin{equation} (\bar{\mathsf{e}}_f)^{o} = f_1(x_4-x_3,x_2-x_1)^{o}, \end{equation}

which follows easily from the definition (8.6).

9.3 Other properties

We mention briefly some directions for further investigation. The proofs of the following facts are trivial yet sometimes lengthy applications of the definitions, basic properties of period polynomials, and the definition of $\{.\,{,}\,.\}$.

Suppose that $f^{(1)}, \ldots , f^{(n)}$ are period polynomials, and let $f^{(i)}_1$ be as defined in § 8.2. Then, generalizing (8.7), we have

(9.5)\begin{equation} \pi^0_{n+2}\pi^0_{n+3} \ldots\pi^0_{4n-1} \pi^0_{4n} \big(\bar{\mathsf{e}}_{f^{(1)}} \circ (\bar{\mathsf{e}}_{f^{(2)}} \circ \cdots (\bar{\mathsf{e}}_{f^{(n-1)}} \circ \bar{\mathsf{e}}_{f^{(n)}}) \cdots )\big) = \prod_{i=1}^n f_1^{(i)}(x_i,x_{i+1}). \end{equation}

Unfortunately, some information about the polynomials $f_i$ is lost in this equation, but one can do better by using the operators $\pi _i^{\mathrm {ev}} = \pi _{x_i}^{\mathrm {ev}}$. For example, one checks that

(9.6)\begin{equation} \pi^{\mathrm{ev}}_1 \pi^{\mathrm{ev}}_5 \pi^0_6 \pi^0_7 \pi^0_8\, (\bar{\mathsf{e}}_{f} \circ \bar{\mathsf{e}}_g )= \big(\pi^{\mathrm{ev}}_1 \pi_4^0 \bar{\mathsf{e}}_f (x_1,x_2,x_3,x_4)\big)\times \big( \pi^{\mathrm{ev}}_5 \pi_6^0 \bar{\mathsf{e}}_g(x_3,x_4,x_5,x_6) \big) \end{equation}

factorizes. Applying the operator $\pi ^2_3$ to this equation gives

\[ \big( \pi^1_3 \pi^{\mathrm{ev}}_1\pi_4^0 \bar{\mathsf{e}}_f (x_1,x_2,x_3,x_4)\big)\times \big( \pi^1_3 \pi^{\mathrm{ev}}_5 \pi_6^0 \bar{\mathsf{e}}_g(x_3,x_4,x_5,x_6) \big)\in \mathbb{Q}[x_1,x_2] \otimes_{\mathbb{Q}} \mathbb{Q}[x_4,x_5] \]

and causes the variables to separate. Next, one checks that

\[ \pi^1_3 \pi^{\mathrm{ev}}_1\pi_4^0 \bar{\mathsf{e}}_f (x_1,x_2,x_3,x_4) = \pi^{\mathrm{ev}}_1 ( \alpha (x_2 -x_1)^{\deg f} + f_0(x_1,x_1+x_2)) \]

for some $\alpha \in \mathbb {Q}$, and it is easy to show that the right-hand side of the previous equation is non-zero and uniquely determines the period polynomial $f$ (using the fact that the involutions $(x_1,x_2)\mapsto (x_1,x_2-2x_1)$ and $(x_1,x_2)\mapsto (-x_1,x_2)$ generate an infinite group). Putting these facts together shows the following result.

Since similar factorization properties to (9.6) hold in higher depths, one might hope to prove, in a similar manner, the conjecture that the Lie subalgebra of $\mathfrak {ls}$ generated by the exceptional elements $\mathsf {e}_f$ is free.

10.1 Interpretation of the Broadhurst–Kreimer conjecture

In the light of the Broadhurst–Kreimer conjecture on the dimensions of the space of multiple zeta values graded by depth (1.2), and Zagier's conjecture which states that the regularized double shuffle relations generate all relations between multiple zeta values, it is natural to rephrase their conjectures, tentatively, in the Lie algebra setting as follows.

Conjecture 4 (Strong Broadhurst–Kreimer and Zagier conjecture)

(10.1)\begin{align} \begin{aligned} H_1(\mathfrak{ls} , \mathbb{Q}) &\cong \mathfrak{ls} _1 \oplus \mathsf{e}(\mathsf{S} ), \\ H_2(\mathfrak{ls} , \mathbb{Q}) &\cong \mathsf{S} , \\ H_i(\mathfrak{ls} , \mathbb{Q}) &= 0, \quad \hbox{for all } i \geq 3 . \end{aligned}\end{align}

This conjecture is the strongest possible conjecture that one could make: as we shall see below, it implies nearly all the remaining open problems concerning relations between (motivic) multiple zeta values. The numerical evidence for this conjecture is substantial [Reference Blümlein, Broadhurst and VermaserenBBV10], but not sufficient to remove all reasonable doubt. More conservatively, and without reference to the double shuffle relations, one could make a weaker reformulation of the Broadhurst–Kreimer conjecture.

Conjecture 5 (Motivic version of the Broadhurst–Kreimer conjecture)

The exceptional elements $\mathsf {e}_f$ are motivic (i.e. $\mathsf {e}(\mathsf {S} ) \subset \mathfrak {dg}^{\mathfrak {m}}$), and

(10.2)\begin{align} \begin{aligned} H_1(\mathfrak{dg}^{\mathfrak{m}}, \mathbb{Q}) &\cong \mathfrak{dg}^{\mathfrak{m}}_1 \oplus \mathsf{e}(\mathsf{S} ), \\ H_2(\mathfrak{dg}^{\mathfrak{m}}, \mathbb{Q}) &\cong \mathsf{S} ,\\ H_i(\mathfrak{dg}^{\mathfrak{m}}, \mathbb{Q}) &= 0, \quad \hbox{for all} \ i \geq 3 . \end{aligned} \end{align}

Since the conjectural generators are totally explicit, it is possible to verify the independence of Lie brackets in the reduced polynomial representation $\bar {\rho }(\mathfrak {g}^{\mathfrak {m}} )$ simply by computing the coefficients of a small number of monomials. In this way, it should be possible to verify (10.2) to much higher weights and depths than is presently known. Note that the Broadhurst–Kreimer conjecture could fail if there existed non-trivial relations between commutators involving several exceptional elements $\mathsf {e}_f$. These would necessarily have weight and depth far beyond the range of present computations.

10.2 Enumeration of dimensions

Let $\mathfrak {h}$ be a Lie algebra over a field $k$, and let $\mathcal {U} \mathfrak {h}$ be its universal enveloping algebra. The homology groups $H_i(\mathfrak {h};k)$ of $\mathfrak {h}$ are defined to be the homology of the following complex:

(10.3)\begin{equation} \cdots\longrightarrow \Lambda^3 \mathfrak{h}\longrightarrow \Lambda^2 \mathfrak{h} \longrightarrow \mathfrak{h} \longrightarrow 0. \end{equation}

Suppose that $\mathfrak {h}$ is bigraded, and finite-dimensional in each bigraded piece. Then $\Lambda ^i \mathfrak {h}, \mathcal {U} \mathfrak {h}, H_i(\mathfrak {h})$ inherit a bigrading. For any bigraded $k$-module $M_{\bullet ,\bullet }$, which is finite-dimensional in every bidegree, define its Poincaré–Hilbert series by

\[ \mathcal{X}_{M} (s,t) = \sum_{m,n\geq 0} \dim_k (M_{m,n}) s^m t^n . \]

Similarly, for a family of such bigraded $k$-modules $M^l$, with $l\geq 0$, let us write

\[ \mathcal{X}_{M^{{\bullet}}} (r,s,t) = \sum_{l,m,n\geq 0} \dim_k (M^l_{m,n}) r^ls^m t^n . \]