NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees
Abstract
Description
Let $K$ be any unital commutative $\bQ$-algebra and $W$ any non-empty subset of $\bN^+$. Let $z=(z_1, ..., z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\kzz$ and $\kttzz$ the formal power series algebras % of $z$ over $K$ and $K[[t]]$, respectively. Let $\cDazz$ $(α\geq 1)$ be the unital algebra generated by the differential operators of $\kzz$ which increase the degree in $z$ by at least $α-1$ and $ \ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\kttzz$ with $o(H_t(z))\geq α$ and $H_{t=0}(z)=0$. First, we study a connection of the \cNcs systems $Ω_{F_t}$ $(F_t\in \ataz)$ (\cite{GTS-I}, \cite{GTS-II}) over the differential operators algebra $\cDazz$ and the \cNcs system $Ω_\bT^W$ (\cite{GTS-IV}) over the Grossman-Larson Hopf algebra $\cH_{GL}^W$ (\cite{GL}, \cite{F1}, \cite{F2}) of $W$-labeled rooted trees. We construct a Hopf algebra homomorphism $\mathcal A_{F_t}: \cH_{GL}^W \to \cDazz$ $(F_t\in \ataz)$ such that $\mathcal A_{F_t}^{\times 5}(Ω_\bT^W) =Ω_{F_t}$. 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 $\cS_{F_t}$ of NCSF's with all $n\geq 1$ and the polynomial automorphisms $F_t=z-H_t(z)$ with $H_t(z)$ homogeneous and the Jacobian matrix $JH_t$ strictly lower triangular can distinguish any two different NCSF's. The graded dualized versions of the main results above are also discussed.
Latex, 35 pages. Some misprints have been corrected
Latex, 35 pages. Some misprints have been corrected