The Goncharov Lie Coalgebra of Observables: Coproduct, Cobracket, and the Decomposition Law

Proof. Write $\Delta(x)=\sum_i a_i\otimes b_i$ with $a_i,b_i$ homogeneous. Counitality gives $\sum_i \varepsilon(a_i) b_i = x$ and $\sum_i a_i\,\varepsilon(b_i)=x$. Because $\mathcal{H}$ is graded and connected, $\varepsilon$ is projection onto $\mathcal{H}_0=\mathbb{Q}\cdot 1$, so the terms with $a_i\in\mathcal{H}_0$ sum to $1\otimes x$ and the terms with $b_i\in\mathcal{H}_0$ sum to $x\otimes 1$. Subtracting these two groups (they overlap only in the term $1\otimes\varepsilon(x)\otimes\cdots$, which vanishes since $\varepsilon(x)=0$ for $x\in\overline{\mathcal{H}}$) leaves precisely the terms with $\operatorname{wt}(a_i)\ge1$ and $\operatorname{wt}(b_i)\ge1$. Hence $\Delta'(x)\in\overline{\mathcal{H}}\otimes\overline{\mathcal{H}}$. ◻

Proof. This is a direct expansion. Coassociativity of $\Delta$ gives $(\Delta\otimes\mathop{\mathrm{id}})\Delta=(\mathop{\mathrm{id}}\otimes\Delta)\Delta$. Writing $\Delta=\Delta'+(\cdot\otimes1)+(1\otimes\cdot)$ on $\overline{\mathcal{H}}$ and expanding both sides, all terms containing a factor $1$ cancel in pairs by counitality, leaving the reduced identity on $\overline{\mathcal{H}}^{\otimes3}$; the surviving terms lie in $\overline{\mathcal{H}}^{\otimes 3}$ by 6 applied slotwise. ◻

Proof. (a) Descent. We must show that for $x,y\in\overline{\mathcal{H}}$ the element $\pi\big((q\otimes q)\Delta'(xy)\big)$ vanishes. Since $\Delta$ is an algebra homomorphism, $$\Delta(xy)=\Delta(x)\,\Delta(y) =\big(x\otimes1+1\otimes x+\Delta'x\big)\big(y\otimes1+1\otimes y+\Delta'y\big).$$ Expanding and subtracting $xy\otimes1+1\otimes xy$ to form $\Delta'(xy)$, the surviving terms are $$\begin{align*} \Delta'(xy) &= \underbrace{x\otimes y+y\otimes x}_{(\ast)} + \underbrace{(x\otimes 1)\Delta'y+(1\otimes x)\Delta'y+(y\otimes1)\Delta'x+(1\otimes y)\Delta'x}_{(\ast\ast)}\\ &\quad + \underbrace{(\Delta'x)(\Delta'y)}_{(\ast\ast\ast)} . \end{align*}$$ We treat the three groups.

Group $(\ast\ast)$. Write $\Delta'y=\sum_j c_j\otimes d_j$ with $c_j,d_j\in\overline{\mathcal{H}}$ (6). Then $(x\otimes1)\Delta'y=\sum_j xc_j\otimes d_j$; its left factors $xc_j$ lie in $\overline{\mathcal{H}}^2$, so $q(xc_j)=0$ and the whole term dies under $q\otimes q$. The term $(1\otimes x)\Delta'y=\sum_j c_j\otimes xd_j$ dies because $xd_j\in\overline{\mathcal{H}}^2$. Symmetrically for the two terms involving $\Delta'x$. Hence $(q\otimes q)$ annihilates all of $(\ast\ast)$.

Group $(\ast\ast\ast)$. Write $\Delta'x=\sum_i a_i\otimes b_i$, $\Delta'y=\sum_j c_j\otimes d_j$; then $(\Delta'x)(\Delta'y)=\sum_{i,j} a_ic_j\otimes b_id_j$ with $a_ic_j,\;b_id_j\in\overline{\mathcal{H}}^2$, so again $q\otimes q$ annihilates it.

Group $(\ast)$. $(q\otimes q)(x\otimes y+y\otimes x)=q(x)\otimes q(y)+q(y)\otimes q(x)$, which is symmetric, hence killed by the antisymmetrizing projection $\pi$.

Therefore $\pi\circ(q\otimes q)\circ\Delta'(xy)=0$, so the composite descends to $\mathcal{L}$. Weight preservation is immediate: $\Delta$, $\Delta'$, $q$, and $\pi$ are all graded of degree $0$, and by 6 every term of $\Delta'$ on $\overline{\mathcal{H}}_n$ has both factors of weight $\ge1$ summing to $n$.

(b) Co-Jacobi. First note that the antisymmetrized map $$\widetilde\delta:=(\mathop{\mathrm{id}}-\tau)\circ(q\otimes q)\circ\Delta'\colon\overline{\mathcal{H}}\longrightarrow\mathcal{L}\otimes\mathcal{L}$$ does descend to $\mathcal{L}$, even though the un-antisymmetrized $(q\otimes q)\circ\Delta'$ does not. Indeed, by the computation in part (a) the only part of $(q\otimes q)\Delta'$ that survives on $\overline{\mathcal{H}}^2$ is the symmetric term $q(x)\otimes q(y)+q(y)\otimes q(x)$ of Group $(\ast)$, and this is annihilated by $(\mathop{\mathrm{id}}-\tau)$. Hence $\widetilde\delta(\overline{\mathcal{H}}^2)=0$, and $\widetilde\delta$ induces a map $\mathcal{L}\to\mathcal{L}\otimes\mathcal{L}$ landing in the antisymmetric tensors, which we identify with $\Lambda^2\mathcal{L}$ via the canonical projection $\pi$ of 9; the induced map is exactly the cobracket $\delta$. (This is precisely the point the un-antisymmetrized descent would have gotten wrong: the raw $(q\otimes q)\Delta'$ carries the symmetric residue and does not factor through $q$.)

For co-Jacobi we lift back to $\overline{\mathcal{H}}$, where $\Delta'$ is coassociative (7): $(\Delta'\otimes\mathop{\mathrm{id}})\Delta'=(\mathop{\mathrm{id}}\otimes\Delta')\Delta'$. The claim $(\mathop{\mathrm{id}}+\sigma+\sigma^2)(\delta\otimes\mathop{\mathrm{id}})\delta=0$ (with $\sigma$ the cyclic permutation of the three factors) is then the classical fact that the antisymmetrization of a coassociative comultiplication is a Lie cobracket — the exact dual, under the pairing with the graded dual, of the statement that the commutator $[a,b]=ab-ba$ of an associative product satisfies the Jacobi identity (Michaelis ). Concretely, expanding $(\delta\otimes\mathop{\mathrm{id}})\delta$ through $\delta=(\mathop{\mathrm{id}}-\tau)\Delta'$ and grouping the resulting terms by the cyclic action, coassociativity of $\Delta'$ makes the six terms cancel in three antisymmetric pairs; the companion formalization verifies this cancellation term-by-term on the cofree model (see the accompanying code). Hence $(\mathcal{L},\delta)$ is a graded Lie coalgebra. ◻

Proof (Goncharov ).. Coassociativity is a combinatorial identity: both $(\Delta\otimes\mathop{\mathrm{id}})\Delta$ and $(\mathop{\mathrm{id}}\otimes\Delta)\Delta$ enumerate pairs of nested subsequences $\{a_{i}\}\subseteq\{a_{j}\}\subseteq\{a_1,\dots,a_N\}$, with the three tensor factors given by the three segments determined by the flag; the two enumerations agree. Compatibility with the shuffle product and gradedness are Goncharov’s Theorem in ; connectedness is $\mathcal{H}_0=\mathbb{Q}$. ◻

Proof. This is a diagram chase. Along the top-right path, bilinearity gives $$(r_\alpha\otimes r_\alpha)(\Delta\mathcal{A}^{\mathfrak{m}}) =(r_\alpha\otimes r_\alpha)\Big(\sum_i\mathcal{A}^{\mathfrak{m}}_{i,1}\otimes\mathcal{A}^{\mathfrak{m}}_{i,2}\Big) =\sum_i r_\alpha(\mathcal{A}^{\mathfrak{m}}_{i,1})\otimes r_\alpha(\mathcal{A}^{\mathfrak{m}}_{i,2}),$$ while along the left-bottom path it equals $\Delta_\alpha(r_\alpha(\mathcal{A}^{\mathfrak{m}}))$. Equating the two paths using the intertwining hypothesis gives [eq:cad]. Existence of $r_\alpha$ as a graded Hopf-algebra homomorphism is the statement that a Tannakian realization functor induces a homomorphism of the associated motivic-Galois Hopf algebras (functoriality of the Tannakian dual, Rung III). ◻

Proof. By 6 every term of $\Delta'(\mathcal{A}^{\mathfrak{m}})$ has both factors of weight $\ge1$ summing to $n$; hence each factor has weight $\le n-1$, and each iteration of $\Delta'$ strictly lowers the weight of every factor. Since weights are positive integers bounded by $n$, after at most $n-1$ further steps each branch reaches a factor on which $\Delta'$ vanishes — by definition a primitive. At weight $1$ this is automatic ($\Delta'$ cannot split $1=p+q$ with $p,q\ge1$, so $\delta|_{\mathcal{L}_1}=0$), but $\Delta'$ may also vanish at higher weight (e.g. $\zeta^{\mathfrak m}(3)$), in which case the branch terminates there. Finiteness of the tree follows since each node has finitely many children (the weight pieces are finite-dimensional over $F$-arithmetic data) and depth is bounded by $n$. ◻

Proof. (b) Weight-one framed mixed Tate motives over $F$ are extensions of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$, classified by $$\mathcal{L}_{G}(F)_1\;\cong\;\mathop{\mathrm{Ext}}^1_{\mathop{\mathrm{MTM}}(F)}(\mathbb{Q}(0),\mathbb{Q}(1))\;\cong\;F^{\times}\otimes\mathbb{Q},$$ the Kummer motives (Rung III): the extension attached to $x\in F^{\times}$ has de Rham class $dt/t$, Betti class a path from $1$ to $x$, and period $\int_1^x dt/t=\log x$. Since weight one cannot split as $p+q$ with $p,q\ge1$, 20 gives $\delta|_{\mathcal{L}_{G}(F)_1}=0$, so $P_1=\mathcal{L}_{G}(F)_1$.

(c) The graded dual $\mathcal{L}_{G}(F)^\vee=\bigoplus_n(\mathcal{L}_{G}(F)_n)^\vee$ is the motivic Galois Lie algebra $\mathfrak{g}=\mathop{\mathrm{Lie}}(\pi_1(\mathop{\mathrm{MTM}}(F)))$ of the pro-unipotent radical of the Tannakian fundamental group of $\mathop{\mathrm{MTM}}(F)$. By Deligne–Goncharov and Borel’s computation of the ranks of $K$-groups of number fields , for a number field $F$ the graded pieces satisfy $$\dim_\mathbb{Q}\big(\mathfrak{g}^{\mathrm{ab}}\big)_n =\dim_\mathbb{Q}\mathop{\mathrm{Ext}}^1_{\mathop{\mathrm{MTM}}(F)}(\mathbb{Q}(0),\mathbb{Q}(n)) =\begin{cases} r_1+r_2 & n\ \text{odd}>1,\\ r_2 & n\ \text{even},\\ r_1+r_2 & n=1\ (\text{i.e. }\mathop{\mathrm{rank}}\,F^{\times}\otimes\mathbb{Q}), \end{cases}$$ and there are no relations forced beyond those visible on generators; equivalently $\mathfrak g$ is a free graded Lie algebra on a set of generators indexed weightwise by these Ext groups. A free graded Lie algebra is generated by its weightwise generators, so its graded dual coalgebra $\mathcal{L}_{G}(F)$ is cogenerated by the corresponding primitives.

(a) A graded connected coalgebra carries the coradical (primitive) filtration $F_0\subseteq F_1\subseteq\cdots$, with $F_0=P$ the primitives and $F_{k}=\{x:\widetilde\delta(x)\in \sum_{i+j=k-1}F_i\otimes F_j\}$. The intersection $\bigcap_k\ker(\widetilde\delta^{[k]})$ of kernels of all iterated coactions is $0$ in a connected graded coalgebra: an element in the intersection has $\widetilde\delta$-image vanishing at every depth, forcing it into $F_0=P$. Hence every element is reconstructed from its iterated cobrackets; by 20 the iteration terminates after at most $n$ steps, and its leaves are primitives — weight-one logarithms $F^{\times}\otimes\mathbb{Q}$ by (b), together with the higher-weight primitives of (c). The union $\bigcup_n\bigoplus_{k\le n}P_k$ of all these primitives is the cogenerating set. ◻

Proof. Define $\{x\}\in\mathcal{L}_{G}(F)_2$ as the class of $\operatorname{Li}^{\mathfrak{m}}_2(x)$. By 24, $\delta\{x\}=(1-x)\wedge x$ in $\Lambda^2(F^{\times}\otimes\mathbb{Q})=\Lambda^2\mathcal{L}_{G}(F)_1$ (21(b)). A direct computation — expand $(1-\xi)\wedge\xi$ for each of the five cross-ratio arguments and use bilinearity and antisymmetry of $\wedge$ — shows the five-term combination $\sum(-1)^i\{[\cdots]\}$ maps to $0$ under $\delta$; this is the well-known verification that the Bloch–Wigner symbol respects the five-term relation at the level of $\Lambda^2$. Hence the five-term combination lies in $\ker\delta=P_2$. Zagier–Bloch–Suslin identify $\ker(\delta|_{\mathcal{L}_{G}(F)_2})$ with the relations defining $B(F)\otimes\mathbb{Q}$, giving the stated identification. ◻

Proof. The monodromy of $\operatorname{per}(\mathcal{A}^{\mathfrak{m}})$ around $s=0$ acts on the Betti realization by the Picard–Lefschetz transformation, whose logarithm is the weight-one nilpotent attached to $\log^{\mathfrak{m}}s$. Under the motivic coaction the monodromy operator is dual to the second tensor slot: $\Delta'$ intertwines analytic continuation with the $H$-comodule structure (Rung III), so the variation $\mathcal{A}^{\mathfrak{m}}\mapsto\mathcal{A}^{\mathfrak{m}}(se^{2\pi i})-\mathcal{A}^{\mathfrak{m}}(s)$ picks out exactly the terms whose second factor is $\log^{\mathfrak{m}}s$, with coefficient $2\pi i=\operatorname{per}(\mathbb{Q}(1))$. Realizing gives [eq:disc]; iterating on the first factor $\mathcal{A}^{\mathfrak{m}}_{i,1}$ (itself a lower-weight amplitude object) yields the multiple discontinuities as iterated first slots. ◻

Proof. (ii) The simple objects of $\mathop{\mathrm{MTM}}(k)$ are only the Tate twists $\mathbb{Q}(n)$, of Hodge type $(-n,-n)$; $H^1(E)$ is simple of rank two with $h^{1,0}=h^{0,1}=1$, hence not Tate. Its period matrix $\left(\begin{smallmatrix}\omega_1&\omega_2\\\eta_1&\eta_2\end{smallmatrix}\right)$ has determinant $2\pi i$ (Legendre relation), and by transcendence results $\omega_1$ is not a $\mathbb{Q}$-linear combination of powers of $\pi$ and multiple zeta values for $E$ without complex multiplication. (i) If $\mathcal{A}^{\mathfrak{m}}$ were a comodule element over $H_{\mathrm{MT}}$, its coaction would express all its anatomy using only Tate generators, contradicting the presence of $H^1(E)$ as a subquotient with genuinely modular ($\mathrm{SL}_2(\mathbb{Z})$) monodromy. (iii) The correct home is the Hopf algebra of iterated integrals of modular forms/functions on $E$ (Levin–Racinet ; Broedel–Duhr–Dulat–Tancredi ); over this larger algebra the comodule hypothesis of 16 is restored. ◻

Proof (conditional/heuristic).. Formally immediate if $G_{c}\subseteq\mathop{\mathrm{Aut}}^\otimes(\omega)$, the Tannakian automorphism group of the fibre functor $\omega=\operatorname{per}$, acts by Hopf automorphisms: it then commutes with the comodule structure by definition, and 16 transports this equivariance through $\mathop{\mathrm{Real}}_\alpha=\operatorname{per}$. The entire content is in the antecedent: a precise definition of $G_{c}$ for a general QFT is open (Brown’s conjecture ); verified instances are the coaction on $\phi^4$ periods up to high loop order (Panzer–Schnetz ), which give status-S evidence for special cases without establishing the general statement. ◻

Proof (Connes–Kreimer ).. $(H_{\mathrm{CK}},\Delta)$, graded by loop number, is connected, hence has a unique antipode making it Hopf. Multiplicativity of $\varphi$ under the Rota–Baxter splitting yields the BPHZ formula as the Birkhoff decomposition of $\varphi$. This is the same formal grammar as the Goncharov coproduct [eq:goncharov]: where Goncharov’s $\Delta$ decomposes the value of an integral (period side), Connes–Kreimer’s $\Delta$ decomposes the graph (combinatorial side); the two are compatible via the period map (Feynman rules $=\operatorname{per}\circ\mathrm{Mot}$), and their compatibility square commutes exactly when the amplitude admits a motivic lift ([S]for mixed Tate cases, e.g. Panzer–Schnetz ). ◻

Proof. Claim (1) is 20. Claim (2) is 21(b),(c) together with 22. Claim (3) is 21(a),(c). Claim (4) combines 16, 17, and 27. The final tame proviso is 29. ◻