REPRESENTATION-THEORY APPROACH TO THE POLYNOMIAL SOLUTION OF Q-DIFFERENCE EQUATIONS - U(Q)(SL(3)) AND BEYOND

A new approach to the theory of polynomial solutions of q-difference e quations is proposed. The approach is based on the representation theo ry of simple Lie algebras G and their q-deformations and is presented here for U(q)(sl(n)). First a q-difference realization of U(q)(sl(n)) in terms of n(n - 1)/2 commuting variables and depending on n - 1 comp lex representation parameters, r(i), is constructed. From this realiza tion lowest weight modules (LWM) are obtained which are studied in det ail for the case n = 3 (the well-known n = 2 case is also recovered). All reducible LWM are found and the polynomial bases of their invarian t irreducible subrepresentations are explicitly given. This also gives a classification of the quasi-exactly solvable operators in the prese nt setting. The invariant subspaces are obtained as solutions of certa in invariant q-difference equations, i.e., these are kernels of invari ant q-difference operators, which are also explicitly given. Such oper ators were not used until now in the theory of polynomial solutions. F inally, the states in all subrepresentations are depicted graphically via the so-called Newton diagrams.