Abstract
The main purpose of the present paper is to give some properties of the Jacobson radical, the Frattini subsystem and c-ideals of a Lie triple system. Some further results concerning the Frattini subsystems of nilpotent and solvable Lie triple systems are obtained. Moreover, we develop initially c-ideals for a Lie triple system and make use of them to give some characterizations of a solvable Lie triple system.
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Bai, R. P., Chen, L. Y. and Meng, D. J., The Frattini subalgebra of n-Lie algebras, Acta. Math. Sin. (English Series), 33, 2007, 847–856.
Barnes, D. W., The Frattini argument for Lie algebras, Math. Z., 133, 1973, 277–283.
Benito, P., Draper, C. and Elduque, A., On some algebras related to simple Lie triple systems, J. Algebra, 219(1), 1999, 234–254.
Chen, L. Y. and Meng, D. J., On the intersection of maximal subalgebras in a Lie superalgebra, Algebra Colloquium, 16(3), 2009, 503–516.
Chen, L. Y., Meng, D. J. and Zhang, Y. Z., The Frattini subalgebra of restricted Lie superalgebras, Acta. Math. Sin. (English Series), 22, 2006, 1343–1356.
Elduque, A., A Frattini theory for malcev algebras, Algebras Groups and Geometries, 1(2), 1984, 247–266.
Elduque, A., On the Frattini subalgebra of a Lie-admissible algebra, Algebras Groups and Geometries, 1(2), 1984, 267–276.
Hopkins, N. C., Nilpotent ideals in Lie and anti-Lie triple systems, J. Alg., 178, 1995, 480–492.
Jacobson, N., Lie Algebras, Interscience Pubglishers, New York-London, 1962.
Jacobson, N., Basic Algebra II, San Francisco, W. H. Freeman, 1980.
Li, D. and Guo, X., The influence of minimal subgroups on the structure of finite groups II, Comm. Alg., 26, 1998, 1913–1922.
Li, D. and Guo, X., The influence of c-normality on the structure of finite groups, J. Pure and Applied Algebra, 150, 2000, 53–60.
Lincoln, M. and Towers, D. A., The Frattini subalgebra of a solvable Lie algebra, Proc. Edin. Math. Soc., 40, 1997, 31–40.
Lister, W. G., A structure theory of Lie triple systems, Trans. Amer. Math. Soc., 72, 1952, 217–242.
Marshall, E. I., The Frattini subalgebra of a Lie algebra, London Math. Soc., 42, 1967, 416–422.
Stitzinger, E. L., On the Frattini subalgebra of a Lie algebras, J. London Math. Soc., 2, 1970, 429–438.
Stitzinger, E. L., The Frattini subalgebras of a class of solvable Lie algebra, Pacific J. Math., 34, 1970, 177–182.
Towers, D. A., A Frattini theory for algebras, Proc. London Math. Soc., 27, 1973, 440–462.
Towers, D. A., On complemented Lie algebras, J. London Math. Soc., 22, 1980, 63–65.
Towers, D. A., C-supplemented subalgebras of Lie algebras, J. Lie Theory, 18, 2008, 717–724.
Towers, D. A., C-ideals of Lie algebras, Comm. Alg., 37, 2009, 4366–4373.
Zhang, Z. X., Chen, L. Y., Liu, W. L. and Bai, X. M., The Frattini subsystem of a Lie triple system, Comm. Alg., 37, 2009, 3750–3759.
Zhang, Z. X., Shi, Y. Q. and Zhao, L. N., Invariant symmetric bilinear forms on Lie triple systems, Comm. Alg., 30(11), 2002, 5563–5573.
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Project supported by the National Natural Science Foundation of China (Nos. 11171055, 11071068, 11126129), the Natural Science Foundation of Jilin Province (No. 201115006), the Scientific Research Foundation for Returned Scholars Ministry of Education of China, the Natural Science Foundation of Zhejiang Province (No.D7080080), Qianjiang Excellence Project (No. 2007R10031), the Innovation Team Foundation of the Department of Education of Zhejiang Province (No.T200924) and the PhD Start-up Foundation of Liaoning University of China (No. 2012002).
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Chen, L., Liu, D. & Xu, X. Some subsystems of a lie triple system closely related to its Frattini subsystem. Chin. Ann. Math. Ser. B 34, 791–800 (2013). https://doi.org/10.1007/s11401-013-0786-8
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DOI: https://doi.org/10.1007/s11401-013-0786-8