Rooted tree maps for multiple \(L\)-values (Q2161343)

The rooted tree maps on the non-commutative polynomial algebra \(\mathbb Q\left<x, y\right>\) in two indeterminates, introduced by the first author [Commun. Number Theory Phys. 13, No. 3, 647--666 (2019; Zbl 1423.16033)], induced a class of relations for multiple zeta values. In this paper, the authors generalize the domain of rooted tree maps to obtain a class of relations for multiple \(L\)-values. We now review the set up. A rooted tree is a finite graph which is connected, has no cycles, and has a distinguished vertex called the root. A product given by the disjoint union of rooted trees is called a rooted forest and we denote by \(\mathcal{H}_{\mathrm{CK}}\) the \(\mathbb Q\)-algebra of forests generated by all trees. We denote by \(\mu_r\) the group of \(r\)-th roots of unity and by \(\mathcal{A}_r\) the non-commutative polynomial ring \(\mathbb Q\left<x, y_s~|~ s \in \mu_r \right>\). For \(f\in \mathcal{H}_{\mathrm{CK}}\), they define the unique \(\mathbb{Q}\)-linear map \(\widetilde{f}\colon\mathcal{A}_r\to\mathcal{A}_r\) satisfying some condition. This unique map is called the rooted tree map. We note that when \(r=1\), the properties of the rooted tree map on \(\mathcal{A}_r\) are the same as these of the rooted tree map for multiple zeta values. Let \[ \mathcal{A}_r^0=\mathbb Q+\sum_{s\in\mu_r}x\mathcal{A}_ry_s+\sum_{s,t\in\mu_r,t\neq1}y_t\mathcal{A}_ry_s. \] and \(z_{k,s}\) denote \(x^{k-1}y_s\in\mathcal{A}_r^0\). Then, every words \(w\in \mathcal{A}_r^0\) can be expressed as \(z_{k_1,s_1}\cdots z_{k_n,s_n}\) for \(n\ge0\), \(k_i\ge1\) and \(s_i\in\mu_r\). Then, the \(\mathbb{Q}\)-linear map \(\mathcal{L}^{\Sha}:\mathcal{A}_r^0\longrightarrow\mathbb{C}\) is defined by \(\mathcal{L}^{\Sha}(1)=1\) and \[ \mathcal{L}^{\Sha}(z_{k_1,s_1}\cdots z_{k_n,s_n})=\sum_{m_1>\cdots>m_n}\left(\prod_{i=1}^{n-1}\frac{s_i^{m_i-m_{i+1}}}{m_i^{k_i}}\right)\frac{s_n^{m_n}}{m_n^{k_n}}, \] which is termed a multiple \(L\)-value of shuffle type. The authors first show that \(\tau\widetilde{f}\tau(\mathcal{A}_r^0)\subset\ker\mathcal{L}^{\Sha}\), where \(\tau\) is the anti-automorphism on \(\mathcal{A}_r\) given by \[ \tau(x)=y_1,\tau(y_1)=x,\text{ and }\tau(y_s)=-y_s\text{ for \(s\in\mu_r\setminus\{1\}\)}. \] Therefore, the map \(\tau\widetilde{f}\tau\) involving a rooted tree map gives the relation between multiple \(L\)-values. They next consider the ladder tree \(\lambda_n\), which has \(n\) vertices among which is just one leaf. They prove that the derivation relation for multiple \(L\)-values coincides with the class of relations induced by the ladder maps \(\widetilde{\lambda_n}\).