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Noncommutative Symmetric Systems over Associative Algebras
This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it
CS systems} over differential
operator algebras in commutative or noncommutative variables ([Z4]); the
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 {\it CS
systems} over associative -algebras. We then prove some results for
CS systems in general; the CS systems over
bialgebras or Hopf algebras; and the universal CS system formed
by the generating functions of certain NCSFs in [GKLLRT]. Finally, we review
some of the main results that will be proved in the followed papers [Z4], [Z6]
and [Z7] as some supporting examples for the general discussions given in this
paper.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
page
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