Bicrossed Products of Hopf algebras, FRT reconstruction theorems and classification of braidings on monoidal categories.

Our first objective is to give a categorical classification of bicrossed products of Hopf algebras. A key role will be played by the concept of matched pair of Hopf algebras and the categorical reformulation of the problem. The result we are targeting at is a Schreier type theorem for bicrossed of Hopf algebras. Moreover, in regard to the theory of Hopf algebras we aim the construction of a general product for Hopf algebras. In Hopf algebras theory various types of products (bicrossed (co)product, biproduct, smash (co)product) have been studied intensively. Our construction will be a general object that contains as special cases all these types of products. Another aim is to construct Hopf Algebras with special properties: finite simensional, separable and quasi-triangular Hopf algebras. Another part of the study of corings. We aim to obtain algebraic properties of the category of representations of a coring and, in particular of an entwinning structure. A milestone in the development of quantum group theory was the introduction of the concept of braided monoidal category. The concrete problem that we want to study is the classification of braidings that can be defined on a given monoidal category.