Reduced Coaction Lie Algebra

• Reduced Coaction Lie Algebra is a Lie subalgebra of the free Lie algebra defined by a specialized coaction equation combined with a skew-symmetry condition.
• It employs extraction operators and the Ihara bracket to ensure closure, interlinking double shuffle relations with the Kashiwara–Vergne framework.
• The algebra plays a crucial role in connecting multiple zeta values, polylogarithmic identities, and deformation quantization with applications in arithmetic and topology.

The reduced coaction Lie algebra, typically denoted as $\mathfrak{rc}_0$, is a distinguished Lie subalgebra of the free Lie algebra $\mathfrak{fr}_k(x_0,x_1)$ over a field $k$ of characteristic zero, defined by a combination of an algebraic (coaction) equation and a skew-symmetry condition. It synthesizes, refines, and generalizes structures emerging from the paper of coaction maps in polylogarithmic, motivic, and Lie-theoretic contexts, and interlaces them with deep arithmetic and algebraic properties, including the double shuffle relations and the Kashiwara–Vergne equations.

1. Definition and Foundational Equations

The defining feature of the reduced coaction Lie algebra is that its elements are solutions $\eta\in\mathfrak{fr}_k(x_0,x_1)$ to the “reduced coaction equation,” with the additional requirement of skew-symmetry: $\begin{align*} \text{(Coaction equation)} \qquad & \mu(\eta) = -r_\eta(x_1) + r_\eta(-x_0) - (\eta)_{x_0} - {}_{x_1}(\eta)\,, \tag{1}\ \text{(Skew-symmetry)} \qquad & \eta(x_0,x_1) = -\eta(x_1,x_0). \tag{2} \end{align*}$ Here, $\mu$ is the reduced coaction (an enhanced version of the Turaev, or necklace, cobracket), and the terms $r_\eta(x)$, $(\eta)_{x_0}$, and ${}_{x_1}(\eta)$ are precise “extraction” and “partial coefficient” operators for Lie series in $x_0,x_1$ (see below for technical definitions).

This combination of constraints singles out a subspace $\mathfrak{rc}_0\subset\mathfrak{fr}_k(x_0,x_1)$, endowed with a canonical Lie algebra structure via the Ihara bracket: $$\{\psi_1, \psi_2\} = d_{\psi_2}(\psi_1) - d_{\psi_1}(\psi_2) - [\psi_1, \psi_2],$$ where $d_{\psi}$ acts as a derivation with $d_{\psi}(x_0)=0$ and $d_{\psi}(x_1) = [x_1, \psi]$.

2. Technical Structure: The Reduced Coaction and Extraction Operators

The reduced coaction $\mu$ is a linear map inspired by the Turaev cobracket, acting on the completed noncommutative power series algebra $A = k\langle\langle x_0, x_1 \rangle\rangle$, and is defined by:

• $\mu(x_0) = \mu(x_1) = 0$
• For a word $w = k_1k_2\ldots k_n$, with $k_i \in \{x_0,x_1\}$,

$$\mu(w) = \sum_{i=1}^{n-1} k_1\ldots k_{i-1}(k_i \odot k_{i+1})k_{i+2}\ldots k_n,$$

where $k\odot k' = \delta_{k,k'}k$.

For a Lie series $\eta$, the operator $r_\eta(x)$ accumulates all coefficients of subwords of the form $x_0^{l+1}x_1$ for $l\ge0$. The operations $(\eta)_{x_0}$ and ${}_{x_1}(\eta)$ extract coefficients multiplying $x_0$ on the right and left, respectively, in the standard expansion.

The equation (1) thus enforces a compatibility of the coaction with boundary extractions and is carefully designed to capture relations between multiple polylogarithmic, motivic and Lie-theoretic structures.

3. Relationship to Double Shuffle and Kashiwara–Vergne Lie Algebras

A central result is the identification and embedding of $\mathfrak{rc}_0$ within the web of key arithmetic Lie algebras:

• Double Shuffle Lie Algebra ($\mathfrak{dmr}_0$): $\mathfrak{dmr}_0$ consists of all Lie series satisfying the (shuffle and stuffle) double shuffle relations. When restricted to skew-symmetric elements, the double shuffle relations are equivalent to certain vanishing conditions for polylogarithmic evaluations of specific defect expressions formed from $\eta$—notably, the vanishing of

$$\alpha(\eta) = \eta_{451} + \eta_{123} - \eta_{432} - \eta_{215} - \eta_{543}$$

under all admissible parameterizations. This yields a rigorous identification:

$$\mathfrak{dmr}_0 \cap \{\text{skew-symmetric}\} \simeq \mathfrak{rc}_0 \text{ (with additional polylogarithmic conditions)}.$$

• Kashiwara–Vergne Lie Algebra ($\mathfrak{krv}_2$): Upon further imposing the “krv1 equation”

$$[x_1, \psi(-x_0-x_1, x_1)] + [x_0, \psi(-x_0-x_1, x_0)] = 0,$$

one can construct a potential $h_\psi$ such that the induced tangential derivation lies in $\mathfrak{krv}_2$. Therefore, there is a natural injection:

$$\mathfrak{rc}_0 + \text{krv1} \longrightarrow \mathfrak{krv}_2\,,$$

creating a direct algebraic connection between the coaction and Kashiwara–Vergne structures.

4. Polylogarithmic and Topological Context

The coaction relation and skew-symmetry stem from polylogarithmic and topological considerations. Skew-symmetry, i.e., $\eta(x_0,x_1) = -\eta(x_1,x_0)$, ensures dihedral symmetries required for structural compatibility with the pentagon equation and the combinatorics underlying polylogarithms and mixed Tate motives. The defect $\alpha(\eta)$ appears as a natural “obstruction” and its evaluation using polylogarithmic techniques identifies the admissibility of solutions to the coaction equation with double shuffle conditions.

The role of the Turaev and Goncharov–Brown coactions is central: the failure of their mutual commutativity is precisely measured by the form of the reduced coaction equation, with the “error term” (arising as a sum of insertion operators $I^\mu$) directly motivating the definition of $\mathfrak{rc}_0$.

5. Explicit Examples and Closure Properties

The Lie algebra $\mathfrak{rc}_0$ is closed under the Ihara bracket. For example, the element

$f_3 = [x_0,[x_0,x_1]] + [x_1,[x_0,x_1]]$

is a skew-symmetric solution: $$\begin{align*} \mu(f_3) &= 0, \ d^L_0(f_3) &= -2x_0x_1 + x_1x_0 - x_1^2, \ d^R_1(f_3) &= x_0^2 + 2x_0x_1 - x_1x_0, \ r_{f_3}(x) &= x^2, \ d^L_0(f_3) + \mu(f_3) + d^R_1(f_3) &= - r_{f_3}(x_1) + r_{f_3}(x_0). \end{align*}$$ The Ihara bracket of such $f_3$ with any other skew-symmetric solution produces a new element of $\mathfrak{rc}_0$.

This closure is proven via direct computation using derivational properties and the defining equations, as in Theorem 1.1 of (Ren, 24 Apr 2025).

6. Algebraic and Arithmetic Significance

The infrastructure of the reduced coaction Lie algebra provides an algebraic “bridge” connecting distinct but deeply related realms:

• Multiple zeta value (MZV) theory through the double shuffle relations.
• Algebraic fundamental groups and the Grothendieck–Teichmüller program, via compatibility with associators and their pentagon relations.
• Deformation quantization and the Kashiwara–Vergne problem, thanks to the realization that elements of $\mathfrak{rc}_0$ with appropriate boundary behavior yield solutions in $\mathfrak{krv}_2$.
• Lie-theoretic and topological quantum field theory constructs, since the reduced coaction generalizes the Turaev necklace bialgebra and encompasses higher genus phenomena through its universal nature.

7. Summary Table: Key Structural Relationships

Algebraic Object	Key Defining Equation(s)	Inclusion/Injection
$\mathfrak{dmr}_0$ (double shuffle)	Double shuffle (shuffle & stuffle), optional skew	$$\mathfrak{dmr}_0 \cap \text{skew} \subset \mathfrak{rc}_0$$
$\mathfrak{rc}_0$ (reduced coaction)	Coaction equation (1), skew condition (2)	—
$\mathfrak{krv}_2$ (Kashiwara–Vergne)	KV equations, tangential derivations	$\mathfrak{rc}_0 +$ krv1 $\rightarrow \mathfrak{krv}_2$

8. Outlook and Functional Role

The reduced coaction Lie algebra $\mathfrak{rc}_0$ integrates polylogarithmic, cohomological, and Lie-theoretic structures within a formal algebraic framework. It provides a universal recipient for the double shuffle Lie algebra under skew-symmetry and a source of solutions for the Kashiwara–Vergne problem upon satisfying additional “krv1” constraints. Its interplay with coaction structures both elucidates the fine arithmetic of multiple zeta values and offers a structural backbone for further developments in motivic Galois theory and transcendental deformation problems.

Key insights provided by $\mathfrak{rc}_0$ include its closure under the Ihara bracket, its explicit coaction equations, and its mediating role in connecting major algebraic objects relevant to arithmetic, geometry, and mathematical physics (Ren, 24 Apr 2025, Howarth et al., 24 Sep 2025).