Noncommutative symmetric systems over associative algebras

This paper is the first of a sequence of papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, A NCS system over the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, NCS systems over differential operator algebras and the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] on the NCS (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134]; the NCS systems over the Grossman–Larson Hopf algebras [R. Grossman, R.G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1) (1989) 184–210. [MR1023294]; L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I, II, Bull. Sci. Math. 126 (3) (2002) 193–239; (4) 249–288. See also math.QA/0105212. [MR1909461]] of labeled rooted trees [W. Zhao, A NCS system over the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136]; as well as their connections and applications to the inversion problem [H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982) 287–330. [MR 83k:14028]; A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, in: Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. [MR1790619]] and specializations of NCSFs [W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, NCS systems over differential operator algebras and the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint]. In this paper, inspired by the seminal work [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218–348. See also hep-th/9407124. [MR1327096]] on NCSFs (noncommutative symmetric functions), we first formulate the notion of NCSsystems   over associative QQ-algebras. We then prove some results for NCS systems in general; the NCS systems over bialgebras or Hopf algebras; and the universal NCS system formed by the generating functions of certain NCSFs in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218–348. See also hep-th/9407124. [MR1327096]]. Finally, we review some of the main results that will be proved in the following papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, A NCS system over the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, NCS systems over differential operator algebras and the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] as some supporting examples for the general discussions given in this paper.