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Noncommutative Symmetric Systems over Associative Algebras · arXivDesk

math/0509133Sep 7, 2005A connection of NCS systems with combinatorial Hopf algebras of M. Aguiar, N. Bergeron and F. Sottile has been added in Remark 2.17. Latex, 32 pages

Noncommutative Symmetric Systems over Associative Algebras

Wenhua Zhao

Combinatorics Quantum Algebra

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Abstract

This paper is the first of a sequence papers ([Z4]--[Z7]) on the ${\mathcal N}$ CS $(\text{noncommutative symmetric})$ systems over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$ CS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion ${\mathcal N}$ CS systems over associative $\mathbb Q$

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This paper is the first of a sequence papers ([Z4]--[Z7]) on the NCS (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables ([Z4]); the NCS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion NCS systems over associative Q-algebras.

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The paper addresses: Noncommutative Symmetric Systems over Associative Algebras.
This paper is the first of a sequence papers ([Z4]--[Z7]) on the NCS (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables ([Z4]); the NCS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]).

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Author comment: A connection of NCS systems with combinatorial Hopf algebras of M. Aguiar, N. Bergeron and F. Sottile has been added in Remark 2.17. Latex, 32 pages