The center of the enveloping algebra of the p-Lie algebras [Figure presented]n [Figure presented]n [Figure presented]n, when p divides n

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Abstract

Let [Figure presented], be a reductive Lie algebra over an algebraically closed field F with charF=p>0. Suppose G satisfies Jantzen's standard assumptions. Then the structure of Z, the center of the enveloping algebra [Figure presented], is described by (the extended) Veldkamp's theorem. We examine here the deviation of Z from this theorem, in case [Figure presented] [Figure presented] or [Figure presented] and p|n. It is shown that Veldkamp's description is valid for [Figure presented]. This implies that Friedlander–Parshall–Donkin decomposition theorem for [Figure presented] holds in case p is good for a semi-simple simply connected G (excluding, if p=2, A1-factors of G). In case [Figure presented] or [Figure presented] we prove a fiber product theorem for a polynomial extension of Z. However Veldkamp's description mostly fails for [Figure presented] and [Figure presented]. In particular Z is not Cohen–Macaulay if n>4, in both cases. Contrary to a result of Kac–Weisfeiler, we show for an odd prime that [Figure presented] and [Figure presented] do not generate [Figure presented]. We also show for [Figure presented] that the codimension of the non-Azumaya locus of Z is at least 2 (if n≥3), and exceeds 2 if n>4. This refutes a conjecture of Brown–Goodearl. We then show that Z is factorial (excluding [Figure presented]), thus confirming a conjecture of Premet–Tange. We also verify Humphreys conjecture on the parametrization of blocks, in case p is good for G.

Original languageEnglish
Pages (from-to)217-290
Number of pages74
JournalJournal of Algebra
Volume504
DOIs
StatePublished - 15 Jun 2018

Bibliographical note

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

  • Enveloping algebra
  • pgl
  • Prime characteristic
  • psl
  • sl

ASJC Scopus subject areas

  • Algebra and Number Theory

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