We give a natural and complete description of Ecalle’s mould–comould formalism within a Hopf–algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes–Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogene...

This article is made up with two parts. In the first part, using a recent result of Schauenburg, one generalizes to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf-Galois objects and Hopf-Galois systems. In this last description, one gives explicitly an inverse for a Hopf-Galois object T together with its generalized antipode. In ...

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their f -vectors. We al...

Actually to any spin model one can associate a vertex model (this is clear from V. Jones’ initial interpretation – in terms of statistical mechanics – of these objects) and the construction of Hopf algebras from complex Hadamard matrices is a particular case of the construction of Hopf algebras from biunitary matrices. The construction of Hopf algebras from biunitary matrices is a particular ca...

We provide a quiver setting for quasi-Hopf algebras, generalizing the Hopf quiver theory. As applications we obtain some general structure theorems, in particular the quasi-Hopf analogue of the Cartier theorem and the Cartier-Gabriel decomposition theorem.

We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products Γ ≀ Sn and the corresponding generalizations of quasi-symmetric functions. Finally, we obtain Hopf algebras of colored parking functions, colored non-crossing partitions and park...

We will define two canonical cohomology theories for Hopf C∗-algebras and for Hopf von Neumann algebras (with coefficients in their comodules). We will then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups will be considered in more details. 1991 AMS Mathematics Classification number: Primary: 46L55, 46L05; Secondary: 43A07, 22D25

All −1-type pointed Hopf algebras with Weyl groups of exceptional type are found. It is proved that every non −1-type pointed Hopf algebra is infinite dimensional. 2000 Mathematics Subject Classification: 16W30, 16G10 keywords: Quiver, Hopf algebra, Weyl group.

We give a universal construction of families of Hopf P-algebras for any Hopf operad P. As a special case, we recover the Connes-Kreimer Hopf algebra of rooted trees.