Bigraded Lie algebras related to MZVs

TL;DR

This paper establishes a deep connection between Goncharov's dihedral Lie coalgebra and Brown's linearized double shuffle Lie algebra, proving they are dual in a bigraded sense and exploring their relations to double shuffle spaces.

Contribution

It constructs explicit isomorphisms linking dihedral Lie coalgebras, double shuffle spaces, and the linearized double shuffle Lie algebra, clarifying their algebraic structures and dualities.

Findings

Proves $D_{ulletullet}$ is the bigraded dual of $rak{ls}$

Establishes isomorphisms between dihedral Lie coalgebras and double shuffle spaces

Shows the equivalence of Lie coalgebra structure and preservation by Ihara bracket

Abstract

We prove that Goncharov's dihedral Lie coalgebra $D_{\bullet\bullet}:={\oplus}_{k\geq m \geq 1} D_{m,k}$ of the trivial group ($\widehat{\mathscr{D}}_{\bullet \bullet}(G)$ of (arxiv:math/0009121) for $G=\{e\}$) is the bigraded dual of Brown's linearized double shuffle Lie algebra $\mathfrak{ls}:={\oplus}_{k\geq m \geq 1}\mathfrak{ls}_m^k\subset \mathbb{Q}\langle x,z \rangle$ whose Lie bracket is the Ihara bracket initially defined over $\mathbb{Q}\langle x,z \rangle$. This by constructing an explicit isomorphism of bigraded Lie coalgebras $D_{\bullet \bullet} \to \mathfrak{ls}^\vee$, where $\mathfrak{ls}^\vee$ is the Lie coalgebra dual in the bigraded sense to $\mathfrak{ls}$. The work leads to the equivalence between the two statements: "$D_{\bullet \bullet}$ is a Lie coalgebra with respect to Goncharov's cobracket formula" and "$ \mathfrak{ls}$ is preserved by the Ihara bracket". We also prove folklore results (that apparently have no written proofs in the literature) stating that for $m \geq 2$: $D_{m,\bullet}:=\oplus_{k\geq m} D_{m,k}$ is graded isomorphic (dual) to Ihara-Kaneko-Zagier's double shuffle space $\mathrm{Dsh}_{m}:=\oplus_{k\geq m} \mathrm{Dsh}_{m}({{k}-m}) \subset \mathbb{Q}[x_1,\dots,x_m]$, and that a given linear map $f_m: \mathbb{Q}\langle x,z \rangle_m \to \mathbb{Q}[x_1,\dots,x_m]$, where $\mathbb{Q}\langle x,z \rangle_m$ is the space linearly generated by monomials of $\mathbb{Q}\langle x,z \rangle$ of degree $m$ with respect to $z$, restricts to a graded isomorphism $\bar{f}_m: \mathfrak{ls}_m:=\oplus_{k\geq m} \mathfrak{ls}_m^k \to \mathrm{Dsh}_{m}$. Here, we establish three explicit compatible isomorphisms $D_{\bullet \bullet} \to \mathfrak{ls}^\vee, D_{m\bullet}\to \mathrm{Dsh}_{m}^\vee$ and $\bar{f}_m: \mathfrak{ls}_m \to \mathrm{Dsh}_{m}$, where $\mathrm{Dsh}_{m}^\vee$ is the graded dual of $\mathrm{Dsh}_{m}$.