Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures, or to get information about the structure of complicated objects by

In this paper we study the representations of two semisimple Hopf algebras related to the symmetric group Sn, namely the bismash products Hn = k Cn #kSn−1 and its dual Jn = k Sn−1 #kCn = (Hn) ∗ , where k is an algebraically closed field of

Abstract. We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H, G, α, β) is deformed using a combinatorial datum (σ, v, r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair ` H,(G, ∗), α ′ , β ′ ´ such that there exist an σ-invariant isomorphism of groups H α⊲⊳β G ∼ = H α ′⊲⊳β ′ (G, ∗). Moreover, if we fix the group H and the automorphism σ ∈ Aut(H) then any σ-invariant isomorphism H α⊲⊳β G ∼ = H α ′ ⊲⊳β ′ G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.