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

Abstract

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 $U({\mathfrak g})$ of finite dimensional k-Lie algebras ${\mathfrak g}$ 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 $U({\mathfrak g})$ for ${\mathfrak g}$ semisimple a conjecture of Humphreys [28] on the block structure of $U({\mathfrak g})$ is confirmed. In the case of $U_{\epsilon}({\mathfrak g})$ for ${\mathfrak g}$ semisimple and $\epsilon$ an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of $U_{\epsilon}({\mathfrak g})$ are determined, and a necessary condition (which may also be sufficient) for a baby Verma $U_{\epsilon}({\mathfrak g})$ -module to be simple is obtained.