The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras

Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras U(g) of finite dimensional k-Lie algebras g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of U(g) for g semisimple a conjecture of Humphreys [28] on the block structure of U(g) is confirmed. In the case of U-is an element of(g) for g semisimple and 6 an odd root of unity we obtain a quantum analogue of a result of Mirkovic and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of U-is an element of(g) are determined, and a necessary condition (which may also be sufficient) for a baby Verma U-is an element of(g)-module to be simple is obtained.