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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Varieties of tori and Cartan subalgebras of restricted Lie algebras

Author: Rolf Farnsteiner
Journal: Trans. Amer. Math. Soc. 356 (2004), 4181-4236
MSC (2000): Primary 17B50
Published electronically: April 16, 2004
MathSciNet review: 2058843
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Abstract: This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus.

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Additional Information

Rolf Farnsteiner
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

PII: S 0002-9947(04)03476-2
Received by editor(s): May 23, 2002
Received by editor(s) in revised form: July 19, 2003
Published electronically: April 16, 2004
Additional Notes: Supported by a Mercator Professorship of the D.F.G
Dedicated: Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday
Article copyright: © Copyright 2004 American Mathematical Society