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

On finite-dimensional Hopf algebras

This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those with abelian group is expected to be completed soon and there is substantial progress in the non-abelian case.Comment: 25 pages. To be presented at the algebra session of ICM 2014. Submitted versio

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

Distinguished Pre-Nichols algebras

We formally define and study the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}(V)$ of a braided vector space of diagonal type $V$ with finite-dimensional Nichols algebra $\mathcal{B}(V)$. The algebra $\widetilde{\mathcal{B}}(V)$ is presented by fewer relations than $\mathcal{B}(V)$, so it is intermediate between the tensor algebra $T(V)$ and $\mathcal{B}(V)$. Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from $\widetilde{\mathcal{B}}(V)$ to $\mathcal{B}(V)$, generalizing results of De Concini and Procesi on quantum groups at roots of unity.Comment: 32 page

Overview of (pro-)Lie group structures on Hopf algebra character groups

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: The Butcher group from numerical analysis and character groups which arise from the Connes--Kreimer theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai