Irreducible Representations Of Two Classes Of Fusion Algebras

Fusion category is an important direction of tensor category,the classification of fusion category is one of the main part of the research on fusion category,Fusion algebra is one of the important ring invariants of the classification of fusion category.This paper focus on the irreducible representations of the near group fusion algebra N(G,n),and the general Haagerup fusion algebra H(G,-1,m)on C.This paper is divided into three parts.In Chapter 1,we mainly introduce the research background and some basic concepts,and summarizes some important conclusions.In Chapter 2,we study the irreducible representation of near group fusion algebra Firstly,we construct a method of extending a representation of G to near group fusion algebra.Secondly,we prove that if the representation of G is irreducible,the extended representation is also irreducible.Finally,we discuss the above construction method,which can construct all the different isomorphic representations of near group fusion algebras using the Wedderburn-Artin theorem.In Chapter 3,we study the irreducible representation of the general Haagerup fusion algebra H(G,-1,m).First,we study the property of the general Haagerup fusion algebra H(G,-1,m),and determine G is abelian under this circumstance.Then,the irreducible representation of H(G,-1,m)which G is cyclic group is given in this paper.We give the irreducible representations of Haagerup fusion algebra H(Z3,-1,1)for example.Besides,when all the non unit elements in G are second order elements,the irreducible representation of H(G,-1,m)is given in this part.Eventually,we give the all irreducible representations of the general Haagerup fusion algebra H(G,-1,m)up to isomorphism,which G is abelian.