The lightcone SUq(2) quantum algebra as dynamical symmetry of the Azbel-Hofstadter problem

Resent investigations on the quantum group symmetry of the Bloch electron in a magnetic field of rational flux by Wiegmann and Zabrodin and Faddeev and Kashaev lead to generalised Bethe-ansatz like equations on high genus algebraic curves. We investigate another manifestation of this symmetry directly at the quantum algebra level, where the Cartesian SUq(2) algebra for values of q, roots of unity, appears on the light cone frame of the Lattice. The Askey-Wilson polynomials and the finite quantum mechanics formalism are shown to be important elements of the problem.