HOPF ALGEBRAS OF ENDOMORPHISMS OF HOPF ALGEBRAS

In the last decades two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves to be 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 the Hopf algebra of symmetric functions Symm. Not least because the right nonconumutative versions are often more beautiful than the commutative ones (not all cluttered up with counting coefficients). NSymm and QSymm are not truly the full noncommutative generalizations. One is maximally noncommutative but cocommutative, while the other is maximally non cocommutative but commutative. There is a common, autodual generalization, the Hopf algebra of permutations of Malvenuto, Poirier, and Reutenauer (MPR). This one is, I think, 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 structures (such as autoduality) come 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 NSymm -> MPR and coalgebra sections of the dual natural projection of Hopf algebras MPR -> QSymm. Several of these will be described explicitly.