Group gradings on simple Lie algebras in positive characteristic
Authors:
Yuri Bahturin, Mikhail Kochetov and Susan Montgomery
Journal:
Proc. Amer. Math. Soc. 137 (2009), 12451254
MSC (2000):
Primary 16W10, 16W50, 17B50, 17B70
Published electronically:
October 20, 2008
MathSciNet review:
2465646
Fulltext PDF
Abstract 
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Additional Information
Abstract: In this paper we describe all gradings by a finite abelian group on the following Lie algebras over an algebraically closed field of characteristic : ( not divisible by ), (, ) and (, even).
 1.
Bahturin, Y.; Goze, M. symmetric spaces, Pacific J. Math., 236 (2008), 121.
 2.
Bahturin, Y.; Giambruno, A. Group gradings on associative algebras with involution, Canad. Math. Bull., 51 (2008), 182194.
 3.
Y.
A. Bahturin, I.
P. Shestakov, and M.
V. Zaicev, Gradings on simple Jordan and Lie algebras, J.
Algebra 283 (2005), no. 2, 849–868. MR 2111225
(2005i:17038), http://dx.doi.org/10.1016/j.jalgebra.2004.10.007
 4.
Yuri
Bahturin and Mikhail
Zaicev, Involutions on graded matrix algebras, J. Algebra
315 (2007), no. 2, 527–540. MR 2351876
(2008h:16027), http://dx.doi.org/10.1016/j.jalgebra.2006.10.046
 5.
Yu.
A. Bahturin and M.
V. Zaicev, Graded algebras and graded identities, Polynomial
identities and combinatorial methods (Pantelleria, 2001), Lecture Notes in
Pure and Appl. Math., vol. 235, Dekker, New York, 2003,
pp. 101–139. MR 2021796
(2005a:16059)
 6.
Yuri
Bahturin and Mikhail
Zaicev, Group gradings on simple Lie algebras of type
“𝐴”, J. Lie Theory 16 (2006),
no. 4, 719–742. MR 2270657
(2007i:17037)
 7.
K.
I. Beidar, M.
Brešar, M.
A. Chebotar, and W.
S. Martindale 3rd, On Herstein’s Lie map conjectures.
III, J. Algebra 249 (2002), no. 1, 59–94.
MR
1887985 (2003c:16042), http://dx.doi.org/10.1006/jabr.2001.9076
 8.
Benkart, G.; Gregory, T.; Premet, A. The recognition theorem for graded Lie algebras in prime characteristic, arXiv:math.RA/0508373 v2 (29 Sep 2005).
 9.
Philip
S. Blau and Wallace
S. Martindale 3rd, Lie isomorphisms in *prime GPI rings with
involution, Taiwanese J. Math. 4 (2000), no. 2,
215–252. MR 1757403
(2001i:16061)
 10.
J.
Dieudonné, Introduction to the theory of formal groups,
Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, 20. MR 0332802
(48 #11128)
 11.
Jens
Carsten Jantzen, Representations of algebraic groups, 2nd ed.,
Mathematical Surveys and Monographs, vol. 107, American Mathematical
Society, Providence, RI, 2003. MR 2015057
(2004h:20061)
 12.
V.
G. Kac, Graded Lie algebras and symmetric spaces, Funkcional.
Anal. i Priložen. 2 (1968), no. 2, 93–94
(Russian). MR
0231944 (38 #270)
 13.
Victor
G. Kac, Infinitedimensional Lie algebras, 2nd ed., Cambridge
University Press, Cambridge, 1985. MR 823672
(87c:17023)
 14.
Susan
Montgomery, Hopf algebras and their actions on rings, CBMS
Regional Conference Series in Mathematics, vol. 82, Published for the
Conference Board of the Mathematical Sciences, Washington, DC; by the
American Mathematical Society, Providence, RI, 1993. MR 1243637
(94i:16019)
 15.
E.
I. Zel′manov, Lie algebras with finite gradation, Mat.
Sb. (N.S.) 124(166) (1984), no. 3, 353–392
(Russian). MR
752226 (86d:17016)
 1.
 Bahturin, Y.; Goze, M. symmetric spaces, Pacific J. Math., 236 (2008), 121.
 2.
 Bahturin, Y.; Giambruno, A. Group gradings on associative algebras with involution, Canad. Math. Bull., 51 (2008), 182194.
 3.
 Bahturin, Y.; Shestakov, I.; Zaicev, M. Gradings on simple Jordan and Lie algebras, J. Algebra, 283 (2005), 849868. MR 2111225 (2005i:17038)
 4.
 Bahturin, Y.; Zaicev, M. Involutions on graded matrix algebras, J. Algebra, 315 (2007), 527540. MR 2351876
 5.
 Bahturin, Y.; Zaicev, M. Graded algebras and graded identities, Polynomial identities and combinatorial methods (Pantelleria, 2001), 101139, Lecture Notes in Pure and Appl. Math., 235, Dekker, New York, 2003. MR 2021796 (2005a:16059)
 6.
 Bahturin, Y.; Zaicev, M. Gradings on simple Lie algebras of type ``A'', J. Lie Theory, 16 (2006), 719742. MR 2270657 (2007i:17037)
 7.
 Beidar, K. I.; Brešar, M.; Chebotar, M. A.; Martindale, W. S., 3rd. On Herstein's Lie map conjectures. III, J. Algebra, 249 (2002), no. 1, 5994. MR 1887985 (2003c:16042)
 8.
 Benkart, G.; Gregory, T.; Premet, A. The recognition theorem for graded Lie algebras in prime characteristic, arXiv:math.RA/0508373 v2 (29 Sep 2005).
 9.
 Blau, P. S.; Martindale, W. S., 3rd. Lie isomorphisms in prime GPI rings with involution, Taiwanese J. Math., 4 (2000), 215252. MR 1757403 (2001i:16061)
 10.
 Dieudonné, J. Introduction to the theory of formal groups. Pure and Applied Mathematics, 20, Marcel Dekker, Inc., New York, 1973. MR 0332802 (48:11128)
 11.
 Jantzen, J. C. Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107, American Math. Soc., Providence, RI, 2003. MR 2015057 (2004h:20061)
 12.
 Kac, V. Graded algebras and symmetric spaces. Funct. Anal. Pril., 2 (1968), 9394. MR 0231944 (38:270)
 13.
 Kac, V. Infinite dimensional Lie algebras, second edition, Cambridge University Press, 1985. MR 823672 (87c:17023)
 14.
 Montgomery, S. Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, American Math. Soc., Providence, RI, 1993. MR 1243637 (94i:16019)
 15.
 Zelmanov, E. Lie algebras with a finite grading. Engl. Transl., Math. USSRSb., 52 (1985), 347385. MR 0752226
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Additional Information
Yuri Bahturin
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada
Email:
yuri@math.mun.ca
Mikhail Kochetov
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada
Email:
mikhail@math.mun.ca
Susan Montgomery
Affiliation:
Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, KAP 108, Los Angeles, California 900892532
Email:
smontgom@math.usc.edu
DOI:
http://dx.doi.org/10.1090/S0002993908096342
PII:
S 00029939(08)096342
Received by editor(s):
July 5, 2007
Received by editor(s) in revised form:
February 8, 2008, and April 21, 2008
Published electronically:
October 20, 2008
Additional Notes:
The first author was partially supported by NSERC grant # 22706004 and by a URP grant, Memorial University of Newfoundland.
The second author was supported by a Startup Grant, Memorial University of Newfoundland.
The third author was supported by NSF grant DMS 0401399.
Communicated by:
Birge HuisgenZimmermann
Article copyright:
© Copyright 2008
American Mathematical Society
