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.
 1.
Nicolas
Bourbaki, Lie groups and Lie algebras. Chapters 1–3,
Elements of Mathematics (Berlin), SpringerVerlag, Berlin, 1989. Translated
from the French; Reprint of the 1975 edition. MR 979493
(89k:17001)
 2.
Jörg
Feldvoss, On the cohomology of restricted Lie algebras, Comm.
Algebra 19 (1991), no. 10, 2865–2906. MR 1129547
(92k:17034), 10.1080/00927879108824299
 3.
Wolfgang
Gaschütz, Nichtabelsche 𝑝Gruppen besitzen
äussere 𝑝Automorphismen, J. Algebra 4
(1966), 1–2 (German). MR 0193144
(33 #1365)
 4.
G.
Hochschild, Cohomology of restricted Lie algebras, Amer. J.
Math. 76 (1954), 555–580. MR 0063361
(16,109a)
 5.
G.
Hochschild, Representations of restricted Lie
algebras of characteristic 𝑝, Proc.
Amer. Math. Soc. 5
(1954), 603–605. MR 0066361
(16,562d), 10.1090/S00029939195400663612
 6.
N.
Jacobson, Restricted Lie algebras of
characteristic 𝑝, Trans. Amer. Math.
Soc. 50 (1941),
15–25. MR
0005118 (3,103g), 10.1090/S00029947194100051180
 7.
N.
Jacobson, A note on automorphisms and
derivations of Lie algebras, Proc. Amer. Math.
Soc. 6 (1955),
281–283. MR 0068532
(16,897e), 10.1090/S00029939195500685329
 8.
Nathan
Jacobson, Lie algebras, Dover Publications, Inc., New York,
1979. Republication of the 1962 original. MR 559927
(80k:17001)
 9.
Eugene
Schenkman, The existence of outer automorphisms
of some nilpotent groups of class 2, Proc.
Amer. Math. Soc. 6
(1955), 6–11. MR 0067111
(16,671c), 10.1090/S00029939195500671117
 10.
Helmut
Strade and Rolf
Farnsteiner, Modular Lie algebras and their representations,
Monographs and Textbooks in Pure and Applied Mathematics, vol. 116,
Marcel Dekker, Inc., New York, 1988. MR 929682
(89h:17021)
 11.
Shigeaki
Tôgô, Outer derivations of Lie
algebras, Trans. Amer. Math. Soc. 128 (1967), 264–276. MR 0213406
(35 #4270), 10.1090/S00029947196702134066
 1.
 N. Bourbaki, Lie Groups and Lie Algebras. Chapters 13 (2 printing), SpringerVerlag, Berlin/Heidelberg/New York/London/Paris/Tokyo, 1989. MR 979493 (89k:17001)
 2.
 J. Feldvoss, On the cohomology of restricted Lie algebras, Comm. Algebra 19 (1991), no. 10, 28652906. MR 1129547 (92k:17034)
 3.
 W. Gaschütz, Nichtabelsche Gruppen besitzen äussere Automorphismen, J. Algebra 4 (1966), no. 1, 12. MR 0193144 (33:1365)
 4.
 G. Hochschild, Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), no. 3, 555580. MR 0063361 (16:109a)
 5.
 G. Hochschild, Representations of restricted Lie algebras of characteristic , Proc. Amer. Math. Soc. 5 (1954), no. 4, 603605. MR 0066361 (16:562d)
 6.
 N. Jacobson, Restricted Lie algebras of characteristic , Trans. Amer. Math. Soc. 50 (1941), no. 1, 1525. MR 0005118 (3:103g)
 7.
 N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6 (1955), no. 2, 281283. MR 0068532 (16:897e)
 8.
 N. Jacobson, Lie Algebras, Dover Publications, Inc., New York, 1979 (unabridged and corrected republication of the original edition from 1962). MR 559927 (80k:17001)
 9.
 E. Schenkman, The existence of outer automorphisms of some nilpotent groups of class , Proc. Amer. Math. Soc. 6 (1955), no. 1, 611; Erratum, Proc. Amer.Math. Soc. 7 (1956), no. 6, 1160. MR 0067111 (16:671c)
 10.
 H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York/Basel, 1988. MR 929682 (89h:17021)
 11.
 S. Tôgô, Outer derivations of Lie algebras, Trans. Amer. Math. Soc. 128 (1967), no. 2, 264276. MR 0213406 (35:4270)
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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
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.
