Bigraded Lie algebras related to multiple zeta values (Q2105135)

Summary: We prove that the dihedral Lie coalgebra \(D_{\bullet \bullet}: = \bigoplus_{k \geq m \geq 1} D_{m, k}\) corresponding to \(\widehat{\mathscr{D}}_{\bullet \bullet} (G)\) of Goncharov (Duke Math. J. 110 (2001), 397-487) for \(G = \{e \}\) is the bigraded dual of the linearized double shuffle Lie algebra \(\mathfrak{ls}: = \bigoplus_{k \geq m \geq 1} \mathfrak{ls}_m^k \subset \mathbb{Q} \langle x, z \rangle\) of Brown (Compos. Math. 157 (2021), 529-572) whose Lie bracket is the Ihara bracket initially defined over \(\mathbb{Q} \langle x, z \rangle\), 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 \(m \geq 2\), \(D_{m, \bullet}: = \bigoplus_{k \geq m} D_{m, k}\) is graded isomorphic (dual) to the double shuffle space \(\mathrm{Dsh}_m: = \bigoplus_{k \geq m} \mathrm{Dsh}_m (k - m) \subset \mathbb{Q} [ x_1, \ldots, x_m ]\) (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307-338), and that the linear map \(f_m: \mathbb{Q} \langle x, z \rangle_m \to \mathbb{Q} [ x_1, \ldots, x_m ]\), 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 \(z\), 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 \(\bar{f}_m: \mathfrak{ls}_m: = \bigoplus_{k \geq m} \mathfrak{ls}_m^k \to \mathrm{Dsh}_m\) (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 \(\mathrm{Dsh}_m\).