The Change Of Bases Of Subalgebra U_A'～+ For Quantum Algebra With Two Ranks

Let v is an indeterminate, A=Z[v,v~-1], A =Q(v) is the fraction field of A. (ajj)n×n is Cartan matrix, uA', is the quantum algebra over A' associated to Cartan matrix (aij)n×n . Let Ei Fi K_i~±1 (1≤i≤n) are generating elements of uA. , UA. is the subalgebra of UA. generated by Ei (1≤i≤n) .This paper has two parts. In the first part two bases of UA. for type A2 are given and the change of bases are discussed. As far as type A2 is concerned, the transition matrix of the change of bases can be decomposed into a product of one upper triangular matrix and one skew- upper triangular matrix. In the product, all elements of the principal diagonal of the upper triangular matrix are 1, all elements of the skew-principal diagonal of the skew-upper triangular matrix are 1. In the second part the change of bases of UA+ for type B2 are discussed .The conclusion of type B2 is more complicated than type A2.