Linear Transformations On The Finite-dimensional Irreducible Sl₂-modules And U_q（sl₂）-modules

Let IF denote an algebraically closed field of characteristic zero.Let X;Y and Z be the equitable basis of Sl₂ over IF and let V denote a finite-dimensional irreducible sl₂-module.Let τ be a linear transformation on V.We show that τ acts on V as αY*,where α is not zero,if and only if（i）τ is raising for the decomposition[X]of V;（ii）τ is lowering for the decomposition[Z]of V.Let ρ be a linear transformation on V.We show that ρ acts on V as a linear combination of 1 and Y if and only if（i） ρ is quasi-lowering for the decomposition[X]of V;（ii） ρ is quasi-raising for the decomposition[Z]of V.Letx;x^-1,z be the equitable generators of the quantum algebra Uq（sl₂）and let Vd,i be the irreducible Uq（sl₂）-module of type 1 and dimension d+1,where d is a nonnegative integer.Let φ be a linear transformation on Vd,1.We show that φ acts on Vd,1 as a linear combination of 1,x y and xy if and only if（i）the matrix representing φ with respect to a standard x-eigebasis is lower bidiagonal;（ii）the matrix representing φ with respect to a standard y-eigenbasis is upper bidiagonal;（iii）the matrix representing φ with respect to a standard z-eigenbasis is tridiagonal.