Some Constructions On Multiplier Hopf Algebras

The goal of this thesis is to generalize the theory of Ore extensions andL-R smash products on Hopf algebras to multiplier Hopf algebras. We focuson constructing structures of multiplier Hopf algebras for Ore extensions andL-R smash products.In Chapter 1, we give a comprehensive survey of the backgrounds andmodern developments of Hopf algebras and multiplier Hopf algebras, andthen introduce some basic contents about multiplier Hopf algebras. At last,we show the main results of this thesis.In Chapter 2, in order to define Hopf-Ore extensions in multiplier Hopfalgebras, we first discuss some properties of Ore extensions of A which isan algebra without identity, extend the endomorphisms and derivations ofA to M(A) and using Ore extensions of M(A) establish comultipications,counits and antipodes. Second, the problem of when there is a multiplierHopf algebra structure on an Ore extension of an algebra without identityis settled. Third, we find a sufficient condition for isomorphism of Hopf-Oreextensions.In Chapter 3, we put L-R smash produts under the frame work of multi-plier Hopf algebras, then obtain a sufficient condition such that a L-R smashproduct admits a mutiplier Hopf algebra structure. Dually, we obtain a suf-ficient condition for a L-R smash coproduct being a multiplier Hopf algebraby using new Sweedler notations.In Chapter 4, a generalized concept of L-R smash produts and somecrossed produts of multiplier Hopf algebras are introduced. We not only showisomophic relations among them as algebras, but also prove the generalizedL-R smash products and generalized diagonal crossed produts are isomorphicas multiplier Hopf algebras.