Michiel Hazewinkel

Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempts to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed.

Quasi-Symmetric Functions

Let Z denote the Leibniz-Hopf algebra, which also turns up as the Solomon descent algebra, and the algebra of noncommutative symmetric functions. As an algebra Z = Z , the free associative algebra over the integers in countably many indeterminates. The co-algebra structure is given by \(\mu({Z_n})=\sum\nolimits_{i = 0}^n{{Z_i}}\otimes{Z_{n-i}}\), Z 0 = 1. Let M be the graded dual of Z. This is the algebra of quasi-symmetric functions. The Ditters conjecture (1972), says that this algebra is a free commutative algebra over the integers. This was proved in [13]. In this paper I give an outline of the proof and discuss a number of consequences and related matters.

Dynamic stochastic models for indexes and thesauri, identification clouds, and information retrieval and storage

The first topic of this partial survey paper is that of the growth of adequate lists of key phrase terms for a given field of science or thesauri for such a field. A very rough ‘taking averages’ deterministic analysis predicts monotonic growth with saturation effects. A much more sophisticated realistic stochatic model confirms that.

Stochastic modelling of scientific terms distribution in publications

In this paper, we address the problem of automatic keywords assignment to scientific publications. The idea to use textual traces learned from training data in a supervised manner to identify appropriate keywords is considered. We introduce the transparent concept of identification cloud as a means to represent the semantics of scientific terms. This concept is mathematically defined by models of scientific terms stochastic distributions over publication texts. Characteristics of models as well as procedures for both non-parametric and parametric estimation of probability distributions are presented.

Multiparameter Quantum Groups and Multiparameter R-Matrices

There exists an (2n) + 1 parameter quantum group deformation of GLn which has been constructed independently by several (groups of) authors. In this note, I give an explicitR-matrix for this multiparameter family. This gives additional information on the nature of this family and facilitates some calculations. This explicitR-matrix satisfies the Yang-Baxter equation. The centre of the paper is Section 3 which describes all solutions of the YBE under the restriction rcdab=0 unlessa, b=c, d. One kind of the most general constituents of these solutions precisely corresponds to the (2n) + 1 parameter quantum group mentioned above. I describe solutions which extend to an enhanced Yang-Baxter operator and, hence, define link invariants. The paper concludes with some preliminary results on these link invariants.

Algebras, rings and modules : non-commutative algebras and rings

Groups and group representations.- Quivers and their representations.- Representations of posets and of finite dimensional algebras.- Frobenius algebras and quasi-Frobenius rings.- Right serial rings.- Tiled orders over discrete valuation rings.- Gorenstein matrices.

Coherence and Uniqueness Theorems for Averaging Processes in Statistical Mechanics

Let S be the set of scalings {n−1:n=1,2,3,...} and let Lz=zZ2, z∈S, be the corresponding set of scaled lattices in R2. In this paper averaging operators are defined for plaquette functions on Lz to plaquette functions on Lz′ for all z′, z∈S, z′=dz, d∈{2,3,4,...}, and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D=1,2,3,... and not only for D=2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.

Probabilistic model for the growth of thesauri

The paper deals with a mathematical model which describes how the collection of key phrases (and key words) evolves as a field of science develops. The experimental material is based on statistical observations on the sets of key phrases which have been assigned to papers in representative major journals in the field in question. Asymptotic properties of the model are considered, as well as estimators for the parameters of the model with particular emphasis on estimators that can indicate at what stage a collection of key phrases can be assumed as complete for the field in question at a given moment in time.

Multiparameter quantum groups and multiparameter

There exists an (2n) + 1 parameter quantum group deformation of GLn which has been constructed independently by several (groups of) authors. In this note, I give an explicitR-matrix for this multiparameter family. This gives additional information on the nature of this family and facilitates some calculations. This explicitR-matrix satisfies the Yang-Baxter equation. The centre of the paper is Section 3 which describes all solutions of the YBE under the restriction rcdab=0 unlessa, b=c, d. One kind of the most general constituents of these solutions precisely corresponds to the (2n) + 1 parameter quantum group mentioned above. I describe solutions which extend to an enhanced Yang-Baxter operator and, hence, define link invariants. The paper concludes with some preliminary results on these link invariants.

R

The tutorial introduction is treated in two papers as follows. 1. Differential Manifolds and Calculus of Manifolds 2. Lie Algebras