Date of Award

Spring 1995

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Donovan Van Osdol

Abstract

This paper develops the theory of crossed product Hopf algebras of pairs of arbitrary Hopf algebras. The theory generalizes the crossed products of (Maj90), the Abelian crossed products of (Hof94) and the crossed product algebras of (BCM86). First, conditions are given on the structures involved that are shown to be equivalent to the existence of the crossed product. Next, a bisimplicial object is found that gives a cohomological description of the conditions. Cleft extensions of pairs of arbitrary Hopf algebras are then defined. These generalize the cleft extension algebras of (Swe68) and the Abelian cleft extensions of (By93); they are equivalent to the extensions of (Hof94), while giving an internal definition of extensions. Finally, the equivalence of crossed products and extensions is proved. Throughout this paper extensive use is made of the relatively new technique of tensor diagrams, without which many of the calculations would be intractable.

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