A relation between Turaev coaction, Goncharov--Brown coaction and the reduced coaction Lie algebra

TL;DR

The paper establishes a precise bridge between the Turaev-type coaction and the Goncharov–Brown coaction via an explicit commuting relation, leading to the reduced coaction equation for Lie series. By introducing the reduced coaction and its skew-symmetric solutions, it defines a new Lie algebra under the Ihara bracket, denoted $\overline{\mathfrak{rc}}_0$, and situates it within the landscape of key Lie algebras associated with multiple zeta values and GT theory. The main technical innovations are the explicit formula $D\circ \widehat{\mu}-(1\otimes \widehat{\mu})\circ D = D^{\mu}$ and the verification that the reduced coaction equation is preserved under the Ihara bracket, establishing the Lie algebra structure. The work connects these algebraic structures to deep objects such as $\mathfrak{grt}_1$, $\mathfrak{dmr}_0$, and $\mathfrak{krv}_2$, contributing to the understanding of the algebraic underpinnings of polylogarithms, mixed Tate motives, and multiple zeta values, with conjectural isomorphisms guiding future research.

Abstract

We present a formula that relates the Turaev coaction and the Goncharov-Brown coaction. Motivated by this relation, we introduce the reduced coaction equation. The skew-symmetric solutions to this equation form a Lie algebra under Ihara bracket.

Theorems & Definitions (19)

• Theorem 1.1
• Definition 1.2: The reduced coaction equation
• Example 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 2.1: Brown2014,Proposition 2.6
• proof
• Theorem 2.2
• proof
• Lemma 3.1