×

Jordan subgroups and orthogonal decompositions. (English. Russian original) Zbl 0704.17002

Algebra Logic 28, No. 4, 248-254 (1989); translation from Algebra Logika 28, No. 4, 382-392 (1989).
Let L be a simple complex Lie algebra, \(G=Aut L^ 0\). It is proved that for every Jordan subgroup of G there exists a uniquely defined transitive orthogonal decomposition of L. Some new decompositions, using this construction, are shown.
Reviewer: V.Ufnarovskij

MSC:

17B20 Simple, semisimple, reductive (super)algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. I. Kostrikin, I. A. Kostrikin, and V. A. Ufnarovskii, ”Orthogonal decompositions of simple Lie algebras,” Dokl. Akad. Nauk SSSR,260, No. 3, 526–530 (1981). · Zbl 0505.17005
[2] A. I. Kostrikin, I. A. Kostrikin, and V. A. Ufnarovskii, ”On decompositions of classical Lie algebras,” Trudy Mat. Inst. im. V. A. Steklova,166, 107–122 (1984). · Zbl 0547.17006
[3] J. G. Thompson, ”A conjugacy theorem for E8,” J. Algebra,38, No. 2, 525–530 (1976). · Zbl 0361.20027 · doi:10.1016/0021-8693(76)90235-0
[4] W. H. Hesselink, ”Special and pure gradings of Lie algebras,” Math. Z.,179, No. 1, 135–149 (1982). · Zbl 0467.17007 · doi:10.1007/BF01173920
[5] D. N. Ivanov, ”Orthogonal decompositions of Lie algebras of type and isotropic bundles,” Usp. Mat. Nauk,42, No. 4, 187–188 (1987).
[6] A. V. Alekseevskii, ”On finite commutative Jordan subgroups of simple complex Lie groups,” Funkts. Anal. Prilozhen.,8, No. 4, 1–4 (1974). · Zbl 0294.47012 · doi:10.1007/BF02028300
[7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985). · Zbl 0568.20001
[8] J. Humphreys, Linear Algebraic Groups, Springer-Verlag (1975). · Zbl 0325.20039
[9] D. Gorenstein, Finite Simple Groups: An Introduction to Their Classification [Russian translation], Mir, Moscow (1985). · Zbl 0672.20010
[10] A. I. Kostrikin, I. A. Kostrikin, and V. A. Ufnarovskii, ”Orthogonal decompositions of simple Lie algebras (type ,” Trudy Mat. Inst. im. V. A. Steklova,158, 105–120 (1981). · Zbl 0526.17003
[11] N. Bourbaki, Lie Groups and Lie Algebras [Russian translation], Chaps. IV–VI, Mir, Moscow (1972). · Zbl 0249.22001
[12] B. Bolt, T. G. Rom, and G. E. Wall, ”On the Clifford collineation, transform and similarity groups. I, II,” J. Austral. Math. Soc.,2, No. 1, 60–79, 80–96 (1961). · Zbl 0097.01702 · doi:10.1017/S1446788700026379
[13] D. Gorenstein, Finite Groups, Harper and Row, New York (1968).
[14] T. A. Springer and R. Steinberg, ”Conjugacy classes of elements,” in: Seminar on Algebraic Groups [Russian translation], Mir, Moscow (1973), pp. 118–262.
[15] H. Luneburg, Translation Planes, Springer-Verlag, New York-Heidelberg-Berlin (1980).
[16] É. B. Vinberg, ”The Weyl group of a graded Lie algebra,” Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 3, 488–526 (1976). · Zbl 0363.20035
[17] R. Griess, ”On a subgroup of order 215. |GL(5,2)| in , the Dempwolff group and ,” J. Algebra,40, No. 1, 271–289 (1976). · Zbl 0348.20011 · doi:10.1016/0021-8693(76)90097-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.