Outer restricted derivations of nilpotent restricted Lie algebras

Feldvoss, Jörg; Siciliano, Salvatore; Weigel, Thomas

doi:10.1090/S0002-9939-2012-11316-4

Outer restricted derivations of nilpotent restricted Lie algebras
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by Jörg Feldvoss, Salvatore Siciliano and Thomas Weigel

Proc. Amer. Math. Soc. 141 (2013), 171-179

DOI: https://doi.org/10.1090/S0002-9939-2012-11316-4

Published electronically: May 17, 2012

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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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