Algebraic dependence of Hamiltonians on the coordinate ring of the quantum group GL_q(n)

On the coordinate ring of GL_q(n), we show that the trace of _qX^m, the q-analogue of the mth power of Xε{lunate}GL_q(n), is represented by the polynomial of tr(_qX^k), 1≤k≤n-1, and det _qX for m≥n by using the quantum Cayley-Hamilton formula. This shows that, if one can take tr(_qX^k), k=1, 2, ..., as commutative Hamiltonians on the coordinate ring of GL_q(n), the number of algebraic independent Hamiltonians is finite. Furthermore we show that the first Hamiltonian tr(_qX) and the second Hamiltonian tr(_qX²) commute with each other. We observe the q-analogue of the Toda molecule by using quantum group symmetry.