|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
25 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees | | Wenhua Zhao
; | | Date: |
7 Sep 2005 | | Subject: | Combinatorics; Quantum Algebra MSC-class: Primary: 05E05, 14R10, 16W30; Secondary: 16W20, 06A11 | math.CO math.QA | | Abstract: | Let $K$ be any unital commutative $Q$-algebra and $W$ any non-empty subset of $N^+$. 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 $cDazz$ $(alphageq 1)$ be the unital algebra generated by the differential operators of $kzz$ which increase the degree in $z$ by at least $alpha-1$ and $ ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $kttzz$ with $o(H_t(z))geq alpha$ and $H_{t=0}(z)=0$. First, we study a connection of the cNcs systems $Omega_{F_t}$ $(F_tin ataz)$ (cite{GTS-I}, cite{GTS-II}) over the differential operators algebra $cDazz$ and the cNcs system $Omega_T^W$ (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 $mathcal A_{F_t}: cH_{GL}^W o cDazz$ $(F_tin ataz)$ such that $mathcal A_{F_t}^{ imes 5}(Omega_T^W) =Omega_{F_t}$. 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 $cT:{mathcal N}Sym o cH_{GL}^{N^+}$ (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 $ngeq 1$ 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. | | Source: | arXiv, math.CO/0509138 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER