Écalle's Dimorphic Transportation

• Écalle’s dimorphic transportation is a structural framework that creates canonical isomorphisms between Lie algebras with commuting symmetries using explicit left/right transport maps.
• It leverages mould theory and the ARI–GARI formalism to systematically construct solutions to non-linear equations from linearized data via adjoint operations and flexion operators.
• Its universal applications underpin multiple zeta value algebra, double-shuffle theory, elliptic extensions, and the Kashiwara–Vergne correspondence, enforcing precise symmetry relations.

Écalle’s dimorphic transportation is a structural framework in mould theory that provides canonical isomorphisms between Lie algebras possessing distinct, commuting symmetries, and systematically constructs solutions to non-linear algebraic or functional equations from linearized data. Its universal features manifest through explicit “transport” (adjoint) maps, which implement “dimorphic” (left/right) actions and organize the passage between linearized and full symmetry regimes. This theory plays a foundational role in multiple zeta value (MZV) algebra, the double-shuffle Lie theory, and the explicit connection between MZV, elliptic multiple zeta values, and the Kashiwara–Vergne (KV) Lie algebra.

1. Fundamentals of Mould Theory and the ARI–GARI Formalism

Écalle’s theory centers on moulds and bimoulds: families of functions or coefficients attached to finite words over specified alphabets, whose values lie in a commutative ring of characteristic zero. The two independent sets of variables—denoted $u_1, u_2, \ldots$ and $v_1, v_2, \ldots$—enable the rigorous construction of objects (moulds) in spaces such as $MU$ (moulds in $u$-variables) and $MV$ (moulds in $v$-variables). The subset $LU \subset MU$ consists of moulds vanishing at length zero.

Composition laws on these spaces are governed by flexion operators such as the generalized product $\mu$ and brackets—specifically Écalle’s preari and ari brackets: $$\mathrm{preari}(A,B)= \mathrm{axit}(A,B)(-) + \mu(A,B)$$

$$\mathrm{ari}(A,B)= \mathrm{preari}(A,B) - \mathrm{preari}(B,A)$$

The length-derivation $\operatorname{der}$ acting as $\operatorname{der}(M)^{(r)}=r\cdot M^{(r)}$ provides a filtration crucial for convergence and normalization of series expansions.

The "gari-dilator" $\operatorname{di}S$ is associated to any $S\in MU$ via the derivation law: $$\operatorname{der}S = \mathrm{preari}(S, \operatorname{di}S)$$ and, at the group level,

$$1+\operatorname{di}S = \mathrm{gari}(\operatorname{invgari}S, \exp(\operatorname{der})S)$$

This dilator is central to the construction of transportation operators.

2. Canonical Dimorphic Transport Maps: Construction and Properties

For a given $S\in MU$, Écalle identifies two (a priori distinct) expansion series for the adjoint action $\operatorname{adari}(S)$ on the Lie algebra $ARI$ (with bracket $\mathrm{ari}$). These series, known as the left and right transports, are defined by:

• Left transport $T_L^S$:

$$\operatorname{adari}(S) = \sum_{s\geq 0} \sum_{r_1,\ldots,r_s\geq 1} \frac{1}{r_1(r_1+r_2)\cdots(r_1+\cdots+r_s)} \circ \ell_{r_1}(\operatorname{di}S) \circ \cdots \circ \ell_{r_s}(\operatorname{di}S)$$

• Right transport $T_R^S$:

$$\operatorname{adari}(S) = \sum_{s\geq 0} \sum_{r_1,\ldots,r_s\geq 1} \frac{1}{r_1(r_1+r_2)\cdots(r_1+\cdots+r_s)} \circ \ell_{r_s}(\operatorname{di}\operatorname{invgari} S) \circ \cdots \circ \ell_{r_1}(\operatorname{di}\operatorname{invgari} S)$$

Here, $\ell_r(A)$ implements left-length insertion of $A$ of size $r$. Both series converge in the length-filtration norm and yield the same endomorphism—$\operatorname{adari}(S)$—on $LU$.

A crucial property is that $T_L^S$ and $T_R^S$ are mutually inverse as endomorphisms of $LU$: $$T_L^S \circ T_R^S = T_R^S \circ T_L^S = \mathrm{id}$$ and satisfy the intertwining conjugation formulae with the derivation: $$[\operatorname{der}, T_L^S] = T_L^S \circ (\operatorname{di}S), \qquad [\operatorname{der}, T_R^S] = - (\operatorname{di}S) \circ T_R^S$$ This “dimorphism” (pair of symmetries) underlies the nomenclature dimorphic transportation and enables canonical passage between solutions with specific symmetries.

3. Lifting Linearized Solutions: The Brown Recursion and Its Geometric Description

Brown’s lifting procedure solves the double-shuffle equations (modulo products) by recursively constructing full solutions from linearized data (Kawamura, 24 Jan 2026). For the space of $\mathbb{Q}$-valued linearized solutions $ls_{\mathbb{Q}}$ and the space $ds_{\mathbb{Q}}$ of full double-shuffle solutions, the lift $\chi_B$ acts as follows: $$\begin{aligned} &\chi_B(f)^{(d)} = f \ &\chi_B(f)^{(<d)} = 0 \ &\chi_B(f)^{(d+r)} = \frac{1}{2r} \sum_{i=1}^r \{ \psi_0^{(i)}, \chi_B(f)^{(d+r-i)} \}_{Ihara} \end{aligned}$$ with $\psi_0$ an explicitly constructed mould.

Crucially, this recursive process can be identified with the left dimorphic transportation by the polar dilator $par$, up to conjugation by swap and anti-involutions: $$\chi_B = \mathrm{swap} \circ \mathrm{anti} \circ T_L^{par} \circ \mathrm{swap} \circ \mathrm{anti}$$ This identification immediately ensures that every lifted solution $\chi_B(f)$ lies in $ds_{\mathbb{Q}}$ for $f \in ls_{\mathbb{Q}}$, and that the procedure realizes the universal mechanism of dimorphic transportation for such non-linear functional equations.

4. Dimorphic Transportation, Elliptic Extensions, and the Double-Shuffle–Kashiwara–Vergne Correspondence

Écalle’s dimorphic transportation generalizes to a wide class of settings where two families of symmetry constraints interact. Notable instances include:

• Elliptic double-shuffle Lie algebra ($ds_{ell}$): Here, the dimorphic transportation arises in constructing an injective Lie map $ds \rightarrow ds_{ell}$, with the map realized explicitly through the adjoint action of the mould $\operatorname{invpal}$ (the inverse of the pal mould), composed with the operator “dar” that introduces an additional weight factor. The resulting objects satisfy $\Delta$-bialternality, encoding the dual symmetries necessary in the elliptic context (Schneps, 2015).
• Kashiwara–Vergne (KV) Lie algebra injection: In the “polar” case governing MZVs, dimorphic transportation via conjugation by the bisymmetral bimould “ess” provides an explicit injection of the (double-shuffle) $dmr_0$ Lie algebra into the $krv_2$ Lie algebra. The method utilizes two adjoint maps corresponding to the two “flexion units,” with each adjoint realizing an explicit Lie isomorphism between dimorphic subalgebras distinguished by alternality and twisted alternality conditions (Kawamura, 25 Sep 2025).

5. Algebraic and Geometric Structure: Fundamental Identities and Filtration

The operation of dimorphic transportation is governed by several key algebraic identities. Among these, the first fundamental identity (Ecalle–Schneps) relates the action of swap and fragari to ganit and crash automorphisms; its corollary, the second fundamental identity, elucidates the transfer of symmetries under adjoint actions.

Filtered by word length, all involved series converge, and the transportation maps uniquely extend solutions specified on low-degree generators. Both the left and right transportations preserve the length filtration, guaranteeing stable passage between linearized and full (nonlinear) solutions.

The bisymmetrality property of certain transport elements (e.g., “ess”, “oss”) is essential for ensuring that both adjoint actions define automorphisms of the full algebraic structures. Proofs of bisymmetrality leverage combinatorial lemmas and finite-difference arguments, with crucial use of Ecalle’s gari-dilator machinery (Kawamura, 25 Sep 2025).

6. Broad Implications and Open Directions

Écalle’s dimorphic transportation has a unifying role in the theory of multiple zeta values, associators, and related motivic and Galois symmetries. In every setting where a Lie algebra with two compatible symmetry constraints (dimorphism) is present—such as double-shuffle/stuffle, bialternality, or push–senary regimes—explicit dimorphic transportation provides canonical isomorphisms and an algorithmic route to extend linearized information to the full non-linear or higher-genus context (Kawamura, 24 Jan 2026, Schneps, 2015, Kawamura, 25 Sep 2025).

Open problems include characterizing, for arbitrary dilators $\psi$, the set of cases where the resulting generalization of Brown’s lift is an isomorphism $ls_{\mathbb{Q}} \to ds_{\mathbb{Q}}$ (the Kimura–Tasaka question). Furthermore, the application of these transportation methods beyond the multiple-zeta and associator paradigms, to settings with ARI-like Lie structures and dilators yet to be explicitly constructed, remains an active avenue of exploration.