The Leonard Pairs Related With Quantum Algebra U_q（sl₂）and Q-tetrahedron Algebra （?）_q

Let K denote an algebraically closed field of characteristic zero.Let V denote a vector space over K with finite positive dimension.A Leofnard pair on V is an ordered pair of linear transformations on V such that,for each of these transformations there exists a basis for V with respect to which the matrix representing that transf’ormation is diagonal and the matrix representing the other transformation is irreducible tridiagonal.In this thesis,we study and solve the following two open questions,which are related with the constructions of’ Leonard pairs,given by professor P.Terwilliger.Problem Ⅰ.Let x^±1,y,z be the equitable generators of the quantum algebra Uq（sl2）and let Vd,1 be the irreducible Uq（sl2）-module of type 1 and dimensiond+1,where d is a nonnegative integer.Find all linear transformations A on Vd,1 such that,on Vd,1 the pair A,x-1 the pair A,y and the pair A,z are all Leonard pairs.Problem Ⅱ.Let x₀₂,x₀₁,x₁₂,x₂₀,x₂₃,x₃₀ be the standard generators of the q-tetrahedron algebra（?）q and let V be a vector space of d+1 dimension,where d is a nonnegative in-teger.Let A,B be a normalized Leonard pair of q-Racah type on V,corresponding the tuple（?）.Set t = （?）-1.Then V can be seen as an evaluation（?）d-moduie with evaluation parametert,denoted by Vd（t）.Show that onVd（t）,the pair （?）,x20,the pair A,x23,the pair A,x3o,the pair B,x02,the pair B,x01 and the pair （?）,x12 are all Leonard pairs.