Article

The primitives of the Hopf algebra of noncommutative symmetric functions

11/2004;
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: In the last decennia two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves important, the Hopf algebra of noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric functions QSymm. It has also become clear that it is important to understand the noncommutative versions of such important structures as Symm the Hopf algebra of symmetric functions. Not least because the right noncommmutative versions are often more beautiful than the commutaive ones (not all cluttered up with counting coefficients). NSymm and QSymm are not truly the full noncommutative generalizations. One is maximally noncommutative but cocommutative, the other is maximally non cocommutative but commutative. There is a common, selfdual generalization, the Hopf algebra of permutations of Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best understood as a Hopf algebra of endomorphisms. In any case, this point of view suggests vast generalizations leading to the Hopf algebras of endomorphisms and word Hopf algebras with which this paper is concerned. This point of view also sheds light on the somewhat mysterious formulas of MPR and on the question where all the extra structure (such as autoduality) comes from. The paper concludes with a few sections on the structure of MPR and the question of algebra retractions of the natural inclusion of Hopf algebras of NSymm into MPR and section of the naural projection of MPR onto QSymm.
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