| The notion of Hopf algebra was introduced in studying algebraic topology and algebraic cohomology.Quantum groups are certain families of non-commutative and non-cocommutative Hopf algebras.The appearance of its concept strongly motivates the development of the theory of Hopf algebras.Quasitriangular Hopf algebra is intro-duced by Drinfeld,and it provides the solution to the Yang-Baxter equation.A large number of significant research results about the theory of Hopf algebras have been pro-duced in the past 30 years.In this thesis,we study the quasitriangular structure and representation rings of a class of the scmisimplc Hopf algebra.Kac and Paljutkin constructed a class of non-commutative and non-cocommutative semisimple Hopf algebra,then Masuoka recon-structed this algebra by lifting method.Ore extension is an important method to con-struct new examples of neither commutative nor cocommutative Hopf algebras.Many interesting quantum algebras can be obtained in this way.We first give the defini-tion of a class of non-commutative and non-cocommutative semisimple Hopf algebra,which can be obtained by the special Ore extension for the given abelian group algebra C[C4ŚC4]with dimension 16 over the complex number field C.It has a sub-Hopf algebra of dimension 8,which is just isomorphic to the unique neither commutative nor cocommutative semisimple 8-dimension Hopf algebra K8.Then the main task is to study the quasitriangular structure of this Hopf algebra.By detailed calculation,all the universal R-matrices for this class of Hopf algebras are given.Finally,we deal with the block and repersentation rings of this Hopf algebra.Its block decomposition and all the irreducible modules are obtained by constructing its primitive central idemptents and their actions on modules.The decomposition of tensor product of all the irreducible modules is given.Furthermore,we describe the representation rings of this Hopf alge-bra by generators and relations explicitly. |
PDF Full Text Request