Outer restricted derivations of nilpotent restricted Lie algebras
Authors:
Jörg Feldvoss, Salvatore Siciliano and Thomas Weigel
Journal:
Proc. Amer. Math. Soc. 141 (2013), 171179
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 finitedimensional 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 onedimensional or it is isomorphic to the threedimensional 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 Lietheoretic analogue of a classical grouptheoretic result of Gaschütz on the existence of power automorphisms of groups. As a consequence we obtain that every finitedimensional nontoral 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
Email:
jfeldvoss@jaguar1.usouthal.edu
Salvatore Siciliano
Affiliation:
Dipartimento di Matematica “E. de Giorgi”, Università del Salento, Via Provinciale LecceArnesano, I73100 Lecce, Italy
Email:
salvatore.siciliano@unisalento.it
Thomas Weigel
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di MilanoBicocca, Via R. Cozzi, No. 53, I20125 Milano, Italy
Email:
thomas.weigel@unimib.it
DOI:
http://dx.doi.org/10.1090/S000299392012113164
PII:
S 00029939(2012)113164
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.
