Hopf algebras on quantum spaces and differential calculus

During the last decades a spectacular development of noncommutative generalizations of differential geometry and Lie group theory has been achieved, and the respective new chapters of mathematical physics are known under the frames of quantum spaces and quantum groups which are the most concrete examples of noncommutative structures. The concept of quantum group is known as a deformation of Hopf algebra corresponding to a Lie group. On the other hand, constructing Hopf algebra for an algebra has recently been an effective method to obtain differential calculus over the algebra.In this thesis we first investigate construction of some noncommutative algebra of polynomial functions by using structures of Hopf algebra on the usual commutative algebra of polynomials. Using the main result of this investigation, we obtain a quantum space with two deformation parameters. A differential calculus is obtained for this quantum space due to its Hopf algebra structure. Thus some deformed derivations and the corresponding Weyl algebra are given. Morever, the Cartan Maurer forms and the relevant vector fields are obtained for the quantum space.In this work we also study how to derive quantum spaces with nonhomogeneous relations from a quantum space with homogeneous relations. Based on our discussions a quantum space with nonhomogeneous relations is obtained from the two-parameter quantum space with homogeneous relations, and a differential calculus and some related results on this space are given.Finally a dual Hopf algebra is given for the two-parameter quantum space with homogeneous relations.Key words: Quantum space, Quantum group, Hopf algebra, Differential calculus, Weyl algebra, Dual Hopf algebra.