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

Abstract 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 SU q (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.