Hopf Bimodules, Coquasibialgebras, and an Exact Sequence of Kac

✍ Scribed by Peter Schauenburg

• Book ID: 102564450
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 427 KB
• Volume: 165
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.2016

✦ Synopsis

Based on the ideas of Tannaka-Kreı ˘n 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 crossproduct Q y K* 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 threecocycle. 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.