Hopf Bimodules, Coquasibialgebras, and an Exact Sequence of Kac

Based on the ideas of Tannaka–Kreın reconstruction, we present a categorical construction that assigns to any cleft Hopf algebra inclusion K⊂H a coquasibialgebra having K* as a Hopf subalgebra. As a special case, the construction gives an intrinsic connection between the bismash product K#Q and the double cross- product QâK* constructed from the same combinatorial data. A cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a Sweedler three-cocycle. Thus our construction assigns to every bicrossproduct (or Hopf algebra extension) of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a Sweedler three-cocycle. Based on this observation we use the construction to prove generalizations of Kac's exact sequence for the group of Hopf algebra extensions of a group algebra by a dual group algebra.