Symmetric Green Ring Of Hopf Algebras

Representation ring was first proposed by J.A.Green when he studied modula representation of a finite group in 1960s, therefore representation ring is known as Green ring, it is the ring structure which given in monoidal category when people study Clebsch-Gordan problem, that is the decomposition of the tensor product of indecomposable modules into a direct sum of indecomposables. In 1964, on the basis of the further development of this work Benson and Carlson apply it to the modular representation theory of the finite gorup, which further promoted the development of Green ring. As an invariant of monoidal categories, the representation rings play a more and more important role in the study of monoidal categories. In recent years, the representation of Hopf algebras has received enormous attention, many experts and scholars also achieved fruitful results.In this master’s thesis, we define an important subring of the Green ring of Hopf algebra, which is symmetric representation ring, and study the algrbra structure.This paper is divided into three chapters. In chapter one we reviewed some basic concepts of bialgebra, Hopf algebra, algebra representation, Green ring of Hopf algebra, dual module, Z- bilinear, and determined generating elements and generating relations of Green ring of group algebra CS3, CA4, and the generator and generating relations of CG where G is a cyclic group with order n.In the second chapter, firstly we describes the definition of the symmetric representation ring and it’s Z-basis, then we prove that when the finite dimensional Hopf algebra is semisimnle. the Z- bilinear form is also associative and non-degenerate on, the symmetric representation ring, and symmetric representation ring is symmetric Z-algebra; Secondly, we compute generating elements and generating relations of symmetric Green ring of group algebra CS3, CA4, and determine the number of generator of group algebra CG and the generator and generating relations of CG when n≤10, where G is a cyclic group with order n.In the third chapter, first we find the dual formula of the indecomposable modules of Taft algebra; next we prove the rank of the symmetric Green ring of Taft algebra, which is as Z-module, and a Z-basis of the Jacobson radical of symmetric representation ring of Taft algebra is also given; Finally we investigate the generators,generating relations of the symmetric Green ring of Taft algebra when n=2,3.