NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees

Let $K$ be any unital commutative $\text{bQ}$-algebra and $W$ any non-empty subset of ${\text{bN}}^{+}$. Let $z=(z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\text{kzz}$ and $\text{kttzz}$ the formal power series algebras % of $z$ over $K$ and $K[[t]]$, respectively. Let $\text{cDazz}$ $(\alpha \ge 1)$ be the unital algebra generated by the differential operators of $\text{kzz}$ which increase the degree in $z$ by at least $\alpha -1$ and $\text{ataz}$ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\text{kttzz}$ with $o(H_t(z))\ge \alpha$ and $H_{t=0}(z)=0$. First, we study a connection of the \cNcs systems ${\mathrm{\Omega }}_{F_t}$ $(F_t\in \text{ataz})$ (\cite{GTS-I}, \cite{GTS-II}) over the differential operators algebra $\text{cDazz}$ and the \cNcs system ${\mathrm{\Omega }}_{\text{bT}}^W$ (\cite{GTS-IV}) over the Grossman-Larson Hopf algebra ${\text{cH}}_{GL}^W$ (\cite{GL}, \cite{F1}, \cite{F2}) of $W$-labeled rooted trees. We construct a Hopf algebra homomorphism ${\mathcal{A}}_{F_t}:{\text{cH}}_{GL}^W\to \text{cDazz}$ $(F_t\in \text{ataz})$ such that $\mathcal A_{F_t}^{\times 5}(\Omega_\bT^W) =\Omega_{F_t}$. Secondly, we generalize the tree expansion formulas for the inverse map (\cite{BCW}, \cite{Wr3}), the D-Log and the formal flow (\cite{WZ}) of $F_t$ in the commutative case to the noncommutative case. Thirdly, we prove the injectivity of the specialization $\cT:{\mathcal N}Sym \to \cH_{GL}^{\bN^+}$ (\cite{GTS-IV}) of NCSF’s (noncommutative symmetric functions) (\cite{G-T}). Finally, we show the family of the specializations ${\text{cS}}_{F_t}$ of NCSF’s with all $n\ge 1$ and the polynomial automorphisms $F_t=z-H_t(z)$ with $H_t(z)$ homogeneous and the Jacobian matrix $J{H}_t$ strictly lower triangular can distinguish any two different NCSF’s. The graded dualized versions of the main results above are also discussed.