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.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

Differential Operator Specializations of Noncommutative Symmetric Functions

Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by \kzz and \kttzz the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $\alpha \geq 1$, let \cDazz 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, for any fixed $\alpha \geq 1$ and F_t\in \ataz, we introduce five sequences of differential operators of \kzz and show that their generating functions form a $\mathcal N$CS (noncommutative symmetric) system [Z4] over the differential algebra \cDazz. Consequently, by the universal property of the $\mathcal N$CS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms \cS_{F_t}: {\mathcal N}Sym \to \cDazz (F_t\in \ataz), which are also grading-preserving when $F_t$ satisfies certain conditions. Note that, the homomorphisms \cS_{F_t} above can also be viewed as specializations of NCSFs by the differential operators of \kzz. Secondly, we show that, in both commutative and noncommutative cases, this family \cS_{F_t} (with all $n\geq 1$ and F_t\in \ataz) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.Comment: Latex, 33 pages. Some mistakes and misprints have been correcte