Outer restricted derivations of nilpotent restricted Lie algebras
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- by Jörg Feldvoss, Salvatore Siciliano and Thomas Weigel PDF
- Proc. Amer. Math. Soc. 141 (2013), 171-179 Request permission
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.References
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Additional Information
- Jörg Feldvoss
- Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688–0002
- MR Author ID: 314800
- Email: jfeldvoss@jaguar1.usouthal.edu
- Salvatore Siciliano
- Affiliation: Dipartimento di Matematica “E. de Giorgi”, Università del Salento, Via Provinciale Lecce-Arnesano, I-73100 Lecce, Italy
- Email: salvatore.siciliano@unisalento.it
- Thomas Weigel
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi, No. 53, I-20125 Milano, Italy
- MR Author ID: 319262
- Email: thomas.weigel@unimib.it
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 171-179
- MSC (2010): Primary 17B30, 17B40, 17B50, 17B55; Secondary 17B05, 17B56
- DOI: https://doi.org/10.1090/S0002-9939-2012-11316-4
- MathSciNet review: 2988720