Bigraded Lie algebras related to MZVs

Abstract

We prove that Goncharov's dihedral Lie coalgebra $D_{\bullet \bullet }:={\oplus }_{k\ge m\ge 1}{D}_{m,k}$ of the trivial group (${\hat{\mathcal{D}}}_{\bullet \bullet }(G)$ of (arxiv:math/0009121) for $G=\{e\}$) is the bigraded dual of Brown's linearized double shuffle Lie algebra $\mathfrak{ls}:={\oplus }_{k\ge m\ge 1}{\mathfrak{ls}}_m^k\subset \mathbb{Q}⟨x,z⟩$ whose Lie bracket is the Ihara bracket initially defined over $\mathbb{Q}⟨x,z⟩$. 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 $m\ge 2$: $D_{m,\bullet }:={\oplus }_{k\ge m}{D}_{m,k}$ is graded isomorphic (dual) to Ihara-Kaneko-Zagier's double shuffle space ${\mathrm{Dsh}}_m:={\oplus }_{k\ge m}{\mathrm{Dsh}}_m(k-m)\subset \mathbb{Q}[x_1,\dots ,x_m]$, and that a given linear map $f_m:\mathbb{Q}⟨x,z{⟩}_m\to \mathbb{Q}[x_1,\dots ,x_m]$, where $\mathbb{Q}⟨x,z{⟩}_m$ is the space linearly generated by monomials of $\mathbb{Q}⟨x,z⟩$ of degree $m$ with respect to $z$, restricts to a graded isomorphism ${\bar{f}}_m:{\mathfrak{ls}}_m:={\oplus }_{k\ge m}{\mathfrak{ls}}_m^k\to {\mathrm{Dsh}}_m$. 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 ${\mathrm{Dsh}}_m$.

26 pages, main result $(b)$ was changed by a more significant result (following from the old version), Exposition and arguments simplified, spaces $V$ and $F$ extended by a line (section 3), pairing $<-,-{>}_{\varphi }$ (section 5) adapted to the extension, the Ihara bracket is studied instead of its restriction and its adjoint is computed (section 6). Old changes: latex issues in the Metadata fixed