Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero. In this note we give a short proof of the fact that a Hopf subalgebra of H is a depth two subalgebra if and only if it is normal Hopf subalgebra.

We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors. We find an example which show that the 8th Kaplansky’s conjecture does not hold for braided Hopf algebras.

We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

We find a generalization of the restricted PBW basis for pointed Hopf algebras over abelian groups constructed by Kharchenko. We obtain a factorization of the Hilbert series for a wide class of graded Hopf algebras. These factors are parametrized by Lyndon words, and they are the Hilbert series of certain graded Hopf algebras.

We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld kG-modules and their corresponding Nichols algebras. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras.

In this paper, we study holomorphic vector bundles on (diagonal) Hopf manifolds. In particular, we give a description of moduli spaces of stable bundles on generic (non-elliptic) Hopf surfaces. We also give a classification of stable rank-2 vector bundles on generic Hopf manifolds of complex dimension greater than two.

Connes and Kreimer have discovered the Hopf algebra structure behind the renormalization of Feynman integrals. We generalize the Hopf algebra to the case of ribbon graphs, i.e. to the case of theories with matrix fields. The Hopf algebra is naturally defined in terms of surfaces corresponding to ribbon graphs. As an example, we discuss the renormalization of Φ4 theory and the 1/N expansion.

We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld kG-modules and their corresponding Nichols algebras. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras.

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatori...

The category of graded, bicommutative Hopf algebras over the prime eld with p elements is an abelian category which is equivalent, by work of Schoeller, to a category of graded modules, known as Dieudonn e modules. Graded ring objects in Hopf algebras are called Hopf rings, and they arise in the study of unstable cohomology operations for extraordinary cohomology theories. The central point of ...