The center of the enveloping algebra of the p-Lie algebras Sl(n,) Pgl(n), psl(n) when p divides n

摘要

Let g = Lie(G), be a reductive Lie algebra over an algebraically closed field F with char F = p 0. Suppose G satisfies Jantzen's standard assumptions. Then the structure of Z, the center of the enveloping algebra U(g), is described by (the extended) Veldkamp's theorem. We examine here the deviation of Z from this theorem, in case g = $In, PgIn or 1:061In and pin. It is shown that Veldkamp's description is valid for pgIn. This implies that Friedlander Parshall Donkin decomposition theorem for F[g](g) holds in case p is good for a semi-simple simply connected G (excluding, if p = 2, Ai-factors of G). In case g = sl(n) or g = sl(n) we prove a fiber product theorem for a polynomial extension of Z. However Veldkamp's description mostly fails for Km and p8In. 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 p that Zp(U(Slp)) and U(Slp)SL0 do not generate Z(U(Slp). We also show for 81n 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 g = 42), thus confirming a conjecture of Premet Tange. We also verify Humphreys conjecture on the parametrization of blocks, in case p is good for G. (C) 2018 Elsevier Inc. All rights reserved.