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Outer restricted derivations of nilpotent restricted Lie algebras

Authors: Jörg Feldvoss, Salvatore Siciliano and Thomas Weigel
Journal: Proc. Amer. Math. Soc. 141 (2013), 171-179
MSC (2010): Primary 17B30, 17B40, 17B50, 17B55; Secondary 17B05, 17B56
Published electronically: May 17, 2012
MathSciNet review: 2988720
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Abstract: In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Tôgô on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschütz on the existence of $ p$-power automorphisms of $ p$-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.

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Additional Information

Jörg Feldvoss
Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688–0002

Salvatore Siciliano
Affiliation: Dipartimento di Matematica “E. de Giorgi”, Università del Salento, Via Provinciale Lecce-Arnesano, I-73100 Lecce, Italy

Thomas Weigel
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi, No. 53, I-20125 Milano, Italy

Keywords: Restricted Lie algebra, nilpotent Lie algebra, $p$-unipotent restricted Lie algebra, torus, Heisenberg algebra, restricted derivation, outer restricted derivation, nilpotent outer restricted derivation, restricted cohomology, $p$-supplement, abelian $p$-ideal, maximal abelian $p$-ideal, maximal $p$-ideal, free module
Received by editor(s): January 28, 2011
Received by editor(s) in revised form: June 16, 2011
Published electronically: May 17, 2012
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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