Separating Ore sets for prime ideals of quantum algebras

Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand-Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac-Moody algebras.