Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative Symmetric Systems over Associative Algebras   This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it
${\mathcal N}$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 {\it ${\mathcal N}$CS
systems} over associative $\mathbb Q$-algebras. We then prove some results for
${\mathcal N}$CS systems in general; the ${\mathcal N}$CS systems over
bialgebras or Hopf algebras; and the universal ${\mathcal N}$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.

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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat