The Classification Of Finite-dimensional Irreducible Modules Of Bannai/Ito Algebra

Let K denote an algebraically closed field with characteristic zero and let α,β,γ be some scalars in K. By the Bannai/Ito algebra denoted as A(α,β,γ), we mean the associative algebra over K with generators x, y, z and relations xy+yx= z+α, yz+zy= x+β, zx+xz= y+γ. In this thesis, we classify the finite-dimensional irreducible A(α,β,γ)-modules up to isomorphism by using the theories of the Leonard pairs and the Leonard triples.This thesis is composed of three chapters and organized as follows:In Chapter 1, we introduce some properties about Leonard pairs, Leonard triples, Leonard pairs of Bannai/Ito type and Leonard triples of Bannai/Ito type.In Chapter 2, we first show that the actions of x,y,z on V are diagonalizable, and any two form a Leonard pair. Then, we obtain the eigenvalues of the actions of x, y, z on V, respectively.In Chapter 3, we classify up to isomorphism the irreducible modules of Bannai/Ito algebra with even and odd dimensions, respectively. And we give all irreducible modules structures by using the theories of Leonard triples of Bannai/Ito type.