Michiel Hazewinkel

Publications (107) View all

• Article: Explicit Polynomial Generators for the Ring of Quasisymmetric Functions over the Integers

  Michiel Hazewinkel
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  ABSTRACT: In (Hazewinkel in Adv. Math. 164:283–300,2001, and CWI preprint,2001) it has been proved that the ring of quasisymmetric functions over the integers is free polynomial. This is a matter that has been of great interest since 1972; for instance because of the role this statement plays in a classification theory for noncommutative formal groups that has been in development since then, see (Ditters in Invent. Math. 17:1–20,1972; in Scholtens’ Thesis, Free Univ. of Amsterdam,1996) and the references in the latter. Meanwhile quasisymmetric functions have found many more applications (see Gel’fand et al. in Adv. Math. 112:218–348, 1995). However, the proofs of the author in the aforementioned papers do not give explicit polynomial generators for QSymm over the integers. In this note I give a (really quite simple) set of polynomial generators for QSymm over the integers. Dans (Hazewinkel dans Adv. Math. 164:283–300,2001, et CWI preprint,2001) il a été démontré que l’anneau de fonctions quasisymétriques est polynomialement libre sur l’anneau de base Z. C’est là une question importante etudiée depuis 1972; par exemple cet énoncé joue un rôle important dans la théorie de la classification des groupes formels noncommutatifs, voir (Ditters dans Invent. Math. 17:1–20, 1972; Scholtens dans Thesis, Free Univ. of Amsterdam, 1996 et les références données). Entretemps, les fonctions quasisymétriques ont reçu beaucoup d’applications (voir Gel’fand et al. dans Adv. Math. 112: 218–348, 1995). Par contre les démonstrations données par l’auteur dans les articles cité plus haut ne fournissent pas des générateurs polynomiaux explicites pour QSymm sur l’anneau des entiers rationels. Dans cette Note nous présentons un ensemble (vraiment très simple) de générateurs polynomiaux pour QSymm sur Z.
  Acta Applicandae Mathematicae 04/2012; 109(1):39-44. · 0.90 Impact Factor
• Article: Hasse-Schmidt derivations and the Hopf algebra of noncommutative symmetric functions

  Michiel Hazewinkel
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  ABSTRACT: Let A be an associative algebra (or any other kind of algebra for that matter). A derivation on A is an endomorphism \del of the underlying Abelian group of A such that \del(ab)=a(\del b)+(\del a)b for all a,b\in A (1.1) A Hasse-Schmidt derivation is a sequence (d_0=id,d_1,d_2,...,d_n,...) of endomorphisms of the underlying Abelian group such that for all n \ge 1 d_n(ab)= \sum_{i=0}^n (d_ia)(d_{n-i}b) (1.2) Note that d_1 is a derivation as defined by (1.1). The individual d_n that occur in a Hasse-Schmidt derivation are also sometimes called higher derivations. A question of some importance is whether Hasse-Schmidt derivations can be written down in terms of polynomials in ordinary derivations. For instance in connection with automatic continuity for Hasse-Schmidt derivations on Banach algebras. Such formulas have been written down by, for instance, Heerema and Mirzavaziri in [5] and [6]. They also will be explicitly given below. It is the purpose of this short note to show that such formulas follow directly from some easy results about the Hopf algebra NSymm of non-commutative symmetric functions. In fact this Hopf algebra constitutes a universal example concerning the matter.
  10/2011;
• Source

  Available from: arxiv.org

  Chapter: Niceness theorems

  Michiel Hazewinkel
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  ABSTRACT: There are many results and constructions in mathematics that are * unreasonably nice *. For instance it appears to be difficult for a set to carry many compatible (algebraic) structures. More precisely, if, say, an algebra carries a compatible *higher* structure the underlying algebra must be very regular. For instance, if an associative unital algebra (over a characteristic zero field) carries a graded connected Hopf algebra structure the underlying algebra is free commutative. There are many such theorems in various different parts of mathematics. This paper gives a number of examples of this phenomenon and of similar phenomena as a preliminary step in starting to examine and try to understand this matter. Besides unreasonably nice constructions and theorems there is also the matter of nice proofs. By this I mainly mean proofs that principally rely on, for instance, the universal properties that define an object, and that do not rely (too much) on calculations. This matter is touched upon in the last section of this paper.
  06/2009: pages 79-125;
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  Available from: arxiv.org

  Article: Witt vectors. Part 1

  Michiel Hazewinkel
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  ABSTRACT: This is the first part of a 2 part survey on the functor of the big and p-adic Witt vectors.
  05/2008;
• Source

  Available from: Ewa Matusiak

  Article: METAPLECTIC OPERATORS ON C n

  Hans G Feichtinger, Michiel Hazewinkel, Norbert Kaiblinger, Ewa Matusiak, Markus Neuhauser
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  ABSTRACT: The metaplectic representation describes a class of automorphism of the Heisen-berg group H = H(G), defined for a locally compact abelian group G. For G = R d , H is the usual Heisenberg group. For the case when G is the finite cyclic group Z n , only par-tial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on C n . We also include applications to Gabor frames.
  The Quarterly Journal of Mathematics 01/2008; · 0.62 Impact Factor