The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Details The Zassenhaus variety of a reductive Lie algebra in positive characteristic The Zassenhaus variety of a reductive Lie algebra in positive characteristic

2008-11-27

arxiv.org/abs/0811.4568v4

Mathematics

Rings and Algebras

Scientific paper

10.1016/j.aim.2009.12.006

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let $Z$ be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g* with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G_2.

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