The Quasitriangular Structures And The Group Of Automorphisms Of Hopf Algebras

The mathematician Hopf introduced the concept of Hopf algebra while studying the topological properties of Lie groups.An algebraic system over the field K called the Hopf algebra if it has both K-algebraic structure and K-coalgebraic structure that satisfy certain compatible conditions.Drinfeld introduced the quasitriangular structures of Hopf algebra.They provide the solutions to the Yang-Baxter equation.On the oth-er hand,the automorphism group of an algebra reflects the symmetry of the algebraic structure.It plays an important role in understanding the algebraic structure.The quasi-triangularies and the automorphism group of Hopf algebra are the interesting fields of Hopf algebras.They are important tools to study the structures of Hopf algebra and the classification of Hopf algebra.In the thesis,we study the quasitriangular structures and automorphism group of Hopf algebra over the field K which contains an n-th primitive roots ? of unit.In detail,we give the definitions of Hopf algebras H2n2 and H4n.By detailed calculation,we give all the universal R-matrices of 2n2-dimension semisimple Hopf algebra H2n2.Further-more,we study the automorphism group of Hopf algebras.Firstly,we give the sets of group-like elements and skew primitive elements of H2n2.Then we construct the auto-morphism group of Hopf algebra H2n2 and prove that it is isomorphic to cyclic group Cn of order n.Secondly,we give the sets of group-like elements and skew primitive elements of H4n.The automorphism group of Hopf algebra H4n is constructed,which is isomorphic to multiplicative group K*=K-{0} if a=0,and it is isomorphic to cyclic group C2 of order 2 if a?0.