Bigraded Lie Algebras Related to Multiple Zeta Values

Abstract

We prove that the dihedral Lie coalgebra $D_{\bullet \bullet }:={⨁}_{k\ge m\ge 1}{D}_{m,k}$ corresponding to ${\hat{\mathcal{D}}}_{\bullet \bullet }(G)$ of Goncharov (Duke Math. J. 110 (2001), 397–487) for $G=\{e\}$ is the bigraded dual of the linearized double shuffle Lie algebra $\mathfrak{ls}:={⨁}_{k\ge m\ge 1}{\mathfrak{ls}}_m^k\subset \mathbb{Q}⟨x,z⟩$ of Brown (Compos. Math. 157 (2021), 529–572) whose Lie bracket is the Ihara bracket initially defined over $\mathbb{Q}⟨x,z⟩$, 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 in Goncharov (Duke Math. J. 110 (2001), 397–487)” and “$\mathfrak{ls}$ is preserved by the Ihara bracket''. We also prove folklore results from Brown (Compos. Math. 157 (2021), 529–572) and Ihara et al. (Compos. Math. 142 (2006), 307–338) (which apparently have no written proofs in the literature) stating that for $m\ge 2$, $D_{m,\bullet }:={⨁}_{k\ge m}{D}_{m,k}$ is graded isomorphic (dual) to the double shuffle space ${\mathrm{Dsh}}_m:={⨁}_{k\ge m}{\mathrm{Dsh}}_m(k-m)\subset \mathbb{Q}[x_1,\dots ,x_m]$ (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307–338), and that the 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 $z$, given by $x^{n_1}z\cdots x^{n_m}z{x}^{n_{m+1}}\mapsto {\delta }_{0,n_{m+1}}{x}_1^{n_1}\cdots x_{n_m}^{n_m}$, with ${\delta }_{a,b}$ the Kronecker delta, restricts to a graded isomorphism ${\bar{f}}_m:{\mathfrak{ls}}_m:={⨁}_{k\ge m}{\mathfrak{ls}}_m^k\to {\mathrm{Dsh}}_m$ (stated in Brown (Compos. Math. 157 (2021), 529–572)). 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 ${\mathrm{Dsh}}_m$.