Group gradings on simple Lie algebras in positive characteristic

Bahturin, Yuri; Kochetov, Mikhail; Montgomery, Susan

doi:10.1090/S0002-9939-08-09634-2

Group gradings on simple Lie algebras in positive characteristic
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by Yuri Bahturin, Mikhail Kochetov and Susan Montgomery

Proc. Amer. Math. Soc. 137 (2009), 1245-1254

DOI: https://doi.org/10.1090/S0002-9939-08-09634-2

Published electronically: October 20, 2008

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Abstract:

In this paper we describe all gradings by a finite abelian group $G$ on the following Lie algebras over an algebraically closed field $F$ of characteristic $p\neq 2$: $\mathfrak {sl}_n(F)$ ($n$ not divisible by $p$), $\mathfrak {so}_n(F)$ ($n\geq 5$, $n\neq 8$) and $\mathfrak {sp}_n(F)$ ($n\geq 6$, $n$ even).

References

Bahturin, Y.; Goze, M. $\mathbb {Z}_2\times \mathbb {Z}_2$-symmetric spaces, Pacific J. Math., 236 (2008), 1–21.
Bahturin, Y.; Giambruno, A. Group gradings on associative algebras with involution, Canad. Math. Bull., 51 (2008), 182–194.
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, DOI 10.1016/j.jalgebra.2004.10.007
Yuri Bahturin and Mikhail Zaicev, Involutions on graded matrix algebras, J. Algebra 315 (2007), no. 2, 527–540. MR 2351876, DOI 10.1016/j.jalgebra.2006.10.046
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
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
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, DOI 10.1006/jabr.2001.9076
Benkart, G.; Gregory, T.; Premet, A. The recognition theorem for graded Lie algebras in prime characteristic, arXiv:math.RA/0508373 v2 (29 Sep 2005).
Philip S. Blau and Wallace S. Martindale 3rd, Lie isomorphisms in ${}^\ast$-prime GPI rings with involution, Taiwanese J. Math. 4 (2000), no. 2, 215–252. MR 1757403, DOI 10.11650/twjm/1500407229
J. Dieudonné, Introduction to the theory of formal groups, Pure and Applied Mathematics, vol. 20, Marcel Dekker, Inc., New York, 1973. MR 0332802
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
V. G. Kac, Graded Lie algebras and symmetric spaces, Funkcional. Anal. i Priložen. 2 (1968), no. 2, 93–94 (Russian). MR 0231944
Victor G. Kac, Infinite-dimensional Lie algebras, 2nd ed., Cambridge University Press, Cambridge, 1985. MR 823672
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, DOI 10.1090/cbms/082
E. I. Zel′manov, Lie algebras with finite gradation, Mat. Sb. (N.S.) 124(166) (1984), no. 3, 353–392 (Russian). MR 752226