الفهرس 
Introduction
Multipliers on algebra.Only 14 pages are availabe for public view
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Abstract
The theory of multiplier Hopf algebra was introduced as a generalization of the usual Hopf algebra to overcome the lack in its duality theory. Also, the Multiplier Hopf group coalgebra is an interesting nontrivial generalization of the Hopf group coalgebra. In this thesis we show that there is a one to one correspondence between the class of multiplier Hopf Gcoalgebras and the class of gcograded multiplier Hopf algebras, and recover the properties of Multiplier Hopf group coalgebra and Hopf group coalgebra using the known results in multiplier Hopf algebra theory. This thesis contains four chapters. In chapter one, we collect the basic definitions and theorems of algebra, coalgebra and bialgebra and use it to deal with the Hopf algebra theory. The definition of quantum groups and some of its basic properties are provided. Also, we present the structure of the group coalgebra and the Hopf group coalgebra and its convolution algebra. . Finally we present the structure of cross product Hopf group coalgebras and the definition of quasitriangular Hopf group coalgebras. In chapter two, we survey the basic characterization of multiplier Hopf algebra and some of its spacial cases like algebraic quantum groups and its duality theory . In chapter three we present the structure of Multiplier Hopf group coalgebra. Also, we show its algebraic structure by constructing the counit and antipode. Chapter four contains the main results of this thesis 1. We define the group cograded multiplier Hopf algebras and find it algebraic characterization. 2. We show that there a one to one correspondence between the class of multiplier Hopf G coalgebras and group cograded multiplier Hopf algebras, and we recover almost of the theorem and proofs using group cograded multiplier Hopf algebras. 3. In the case of group cograded multiplier Hopf algebras with finite dimensional components we show that the definitions of Hopf group coalgebra by Tureav and Virelizier coincide. We recover results obtained by Virelizier. We also give an explicit formula for the integrals.