Hopf algebras of endomorphisms of Hopf algebras

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.