The category of group-graded modules over an abelian group G is a monoidal category. For any bicharacter of G this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have n-ary multiplications between various graded components. They possess universal enveloping algeb...

William M. Singer’s theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p3. The theory shows that there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of which is the Hopf algebra corre...

The main result in this paper states that every strongly graded bialgebra whose component of grade 1 is a finite-dimensional Hopf algebra is itself a Hopf algebra. This fact is used to obtain a group cohomology classification of strongly graded Hopf algebras, with 1-component of finite dimension, from known results on strongly graded bialgebras.  2002 Elsevier Science (USA)

In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H

In an attempt to study the zero divisors in infinite Hopf algebras, we study two non-trivial examples of non-group ring infinite Hopf algebras and show that a variant of Kaplansky’s classical zero divisor conjecture holds for these two Hopf algebras.

We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.

The classification of finite-dimensional pointed Hopf algebras with group S3 was finished in [AHS]: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finit...

We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra H associated to the factorization of a finite group into two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that self-duality structures for the bicrossproduct ...