Almost Triangular Hopf Algebra And Hopf Algebras, Galois Extension

This thesis concludes two main parts. First, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra (H, R) with R²¹R ∈ C(H (?) H) (we term this type of Hopf algebras as almost-triangular) over an algebraically closed field k. We write by B the vector space generated by the left tensorand of R²¹ R. Then B is a sub-Hopf algebra of H. We proved that when dim B is odd, H has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [EG00]; when dim B is even, H is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.Let A be a bialgebra over a field k and a cosemisimple Hopf algebra H coacts on A, A is Galois over its invariants. In the second part of this thesis we prove that if the Hopf Galois extension is cocentral, and the invariants is a Hopf algebra then A is itself a Hopf algebra.