Mathematische Zeitschrift

, Volume 238, Issue 4, pp 733–779

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

  • Kenneth A. Brown
  • Iain Gordon
Original article

DOI: 10.1007/s002090100274

Cite this article as:
Brown, K. & Gordon, I. Math Z (2001) 238: 733. doi:10.1007/s002090100274

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Kenneth A. Brown
    • 1
  • Iain Gordon
    • 1
  1. 1.Mathematics Department, University of Glasgow, Glasgow G12 8QW, UK (e-mail: kab@maths.gla.ac.uk; ig@maths.gla.ac.uk)GB