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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.