Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations an...

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...

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B ⊆ A and also to the equivalence of the category of relative Hopf...

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.

For a Hopf algebra A, we define the structures of differential complexes on two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an exterior extension of the dual algebra A. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on A. The first differential complex is an analog of the de Rham complex. In t...

The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal enveloping algebra of a finite dimensional Lie algebra g with a finite subgroup G of automorphisms of g is Calabi-Yau if and only if the universal en...

We introduce the concept of extended Hopf algebras and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structu...

Let H be a finite dimensional Hopf C*-algebra, H1 Hopf*-subalgebra of H. This paper focuses on the observable algebra AH1 determined by in nonequilibrium spin models, which there is copy each lattice site, and ? link, where denotes dual Furthermore, using iterated twisted tensor product +*-algebras, one can prove that algebraAH1 *-isomorphic to C*-inductive limit ... o .