The primitives of the Hopf algebra of noncommutative symmetric functions

11/2004; DOI: 10.11606/issn.2316-9028.v1i2p175-202
Source: arXiv

ABSTRACT Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series over it. This gives another proof of the theorem that the algebra of quasi-symmetric functions is free over the integers.

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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.
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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 11/2004; 109(1):39-44. · 0.99 Impact Factor

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