NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees
| dc.creator | Zhao, Wenhua | |
| dc.date | 2005-09-07 | |
| dc.date | 2005-09-09 | |
| dc.date.accessioned | 2026-07-07T05:23:00Z | |
| dc.date.available | 2026-07-07T05:23:00Z | |
| dc.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. | |
| dc.description | Latex, 35 pages. Some misprints have been corrected | |
| dc.identifier | https://arxiv.org/abs/math/0509138 | |
| dc.identifier | http://arxiv.org/abs/math/0509138 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76275 | |
| dc.subject | Combinatorics | |
| dc.subject | Quantum Algebra | |
| dc.subject | Primary: 05E05, 14R10, 16W30; Secondary: 16W20, 06A11 | |
| dc.title | NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees | |
| dc.type | text |