Construction Of A_R-bialgebra And Its Algebra*-structure

This paper summarizes some existing concepts and properties of algebra, coalgebra, bialgebra, Hopf algebra and gives some review about algebra AR constructed by a special class of Hayashi’s R-matrix. As mentioned above, we define a*-structure map on AR from some theorems and properties which AR satisfies, i. e. we define a conjugate linear map satisfying anti-homomorphism property on AR such that it becomes a*-algebra. Also we prove that the*-structure of algebra AR is consistent. Finally, through substantial computations, arguments and inductions, we get the concrete structure of the*-operation on the linear basis of AR.This paper is divided into three chapters. The first chapter mainly covers the background of the research and some basic but necessary knowledge. The second chapter mainly introduces algebra, coalgebra, and bialgebra and Hopf algebra, and also gives some basic definitions, key propositions and typical examples. The third chapter introduces the physics background of Yang-Baxter equation and gives the specific expressions. This chapter also states that R-matrix is a solution of the Yang-Baxter equation. Based on this, we consider the FRT algebra AR constructed by a special kind of Hayashi’s R-matrix. By defining a*-structure on AR, we make it a*-algebra. At last, the chapter gives the specific representation of*operation on the linear basis elements of AR.