Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E with 1 E ∈ H and the multiplication map A ⊗ H → E is bijective. The tool we use is a new product, we call it the unified product, in the construction of which A and H are connected by three coalgebra maps: two actions and a generalized cocycle. Both the crossed product of an Hopf algebra acting on an algebra and the bicrossed product of two Hopf algebras are special cases of the unified product. A Hopf algebra E factorizes through A and H if and only if E is isomorphic to a unified product of A and H. All such Hopf algebras E are classified up to an isomorphism that stabilizes A and H by a Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as defined by Bichon and Kassel plays the key role in the classification theorem.

Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object H 2

Abstract. Let g be a Lie algebra, E a vector space containing g as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on E such that g is a Lie subalgebra of E. A general product, called the unified product, is introduced as a tool for our approach. Let V be a complement of g in E: the unified product g ♮ V is associated to a system (⊳, ⊲, f, {−, −}) consisting of two actions ⊳ and ⊲, a generalized cocycle f and a twisted Jacobi bracket {−, −} on V . There exists a Lie algebra structure [−, −] on E containing g as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras (E, [−, −]) ∼ = g ♮ V . All such Lie algebra structures on E are classified by two cohomological type objects which are explicitly constructed. The first one H 2 g (V, g) will classify all Lie algebra structures on E up to an isomorphism that stabilizes g while the second object H 2 (V, g) provides the classification from the view point of the extension problem. Several examples that compute both classifying objects H 2 g (V, g) and H 2 (V, g) are worked out in detail in the case of flag extending structures.

Abstract. We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family of coalgebras (resp. bialgebras, Hopf algebras). The equalizers of two morphisms of coalgebras (resp. bialgebras, Hopf algebras) are also described explicitly. As a consequence the categories of coalgebras, bialgebras and Hopf algebras are shown to be complete, and a complete description for limits in the above categories is given.