Leonard Triples And Their Pseudo Intertwiners

Let F denote an algebraically closed field of characteristic zero.Let V denote a non-zero vector space over F with finite positive dimension.Let A:V→V,A*:V→V and Aε:V→V be linear transformations on V.The ordering triple(A,A*,Aε)is called a Leonard triple on V,if for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrices representing the other transformations are irreducible tridiagonal.Let(A,A*,Aε)denote a Leonard triple on V.By the pseudo-intertwiners of(A,A*,Aε),we mean three invertible linear transformations W,W*,Wε on V,which satisfy the conditions:(1).A commutes with W and W-1A*W-Aε;(2).A*commutes with W*and(W*)-1AεW*-A;(3).Aεcommutes with Wε and(Wε)-1AWε-A*.In this thesis,we mainly study the pseudo-intertwiners of the Leonard triples of Bannai/Ito type and those of the Leonard triples of Krawtchouk type.Let(A,A*,Aε)be a Leonard triple that is of Bannai/Ito type or Krawtchouk type.We first construct the pseudo-intertwiners of(A,A*,Aε).Then we express the elements A,A*and Aεas the polynomials in A,A*and Aε,respectively,where A=W-1A*W-Aε,A*=(W*)-1AεW*-A,Aε=(Wε)-1AWε-A*.