Restricted Lie algebras in which every restricted subalgebra is an ideal

Siciliano, Salvatore

doi:10.1090/S0002-9939-09-09780-9

Restricted Lie algebras in which every restricted subalgebra is an ideal

Author: Salvatore Siciliano
Journal: Proc. Amer. Math. Soc. 137 (2009), 2817-2823
MSC (2000): Primary 17B05, 17B50
DOI: https://doi.org/10.1090/S0002-9939-09-09780-9
Published electronically: April 10, 2009
MathSciNet review: 2506437
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Abstract: We characterize restricted Lie algebras over perfect fields all of whose restricted subalgebras are ideals.

1. R. K. Amayo, Lie algebras in which every 𝑛-generator subalgebra is an 𝑛-step subideal, J. Algebra 31 (1974), 517–542. MR 0347919, https://doi.org/10.1016/0021-8693(74)90131-8
2. Ralph K. Amayo and Ian Stewart, Infinite-dimensional Lie algebras, Noordhoff International Publishing, Leyden, 1974. MR 0396708
3. R. Baer, Situation der Untergruppen und Struktur der Gruppe, S. B. Heidelberg. Akad. Wiss. 2 (1933), 12-17.
4. Yuri Bahturin and Alexander Olshanskii, Large restricted Lie algebras, J. Algebra 310 (2007), no. 1, 413–427. MR 2307801, https://doi.org/10.1016/j.jalgebra.2006.12.018
5. Yiftach Barnea and I. M. Isaacs, Lie algebras with few centralizer dimensions, J. Algebra 259 (2003), no. 1, 284–299. MR 1953720, https://doi.org/10.1016/S0021-8693(02)00558-6
6. G. M. Benkart and I. M. Isaacs, Lie algebras with nilpotent centralizers, Canad. J. Math. 31 (1979), no. 5, 929–941. MR 546949, https://doi.org/10.4153/CJM-1979-088-8
7. Kevin Bowman, David A. Towers, and Vicente R. Varea, On flags and maximal chains of lower modular subalgebras of Lie algebras, J. Lie Theory 17 (2007), no. 3, 605–616. MR 2352003
8. R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann. 48 (1897), no. 4, 548–561 (German). MR 1510943, https://doi.org/10.1007/BF01447922
9. Masanobu Honda, On Lie algebras in which every subalgebra is a subideal, Hiroshima Math. J. 18 (1988), no. 3, 655–660. MR 991251
10. Mark Lincoln and David Towers, Frattini theory for restricted Lie algebras, Arch. Math. (Basel) 45 (1985), no. 5, 451–457. MR 815984, https://doi.org/10.1007/BF01195370
11. Shao-xue Liu, On algebras in which every subalgebra is an ideal, Chinese Math.–Acta 5 (1964), 571–577. MR 0172899
12. D. L. Outcalt, Power-associative algebras in which every subalgebra is an ideal, Pacific J. Math. 20 (1967), 481–485. MR 0207784
13. D. M. Riley, Restricted Lie algebras of maximal class, Bull. Austral. Math. Soc. 59 (1999), no. 2, 217–223. MR 1680807, https://doi.org/10.1017/S0004972700032834
14. D. M. Riley and J. F. Semple, The coclass of a restricted Lie algebra, Bull. London Math. Soc. 26 (1994), no. 5, 431–437. MR 1308359, https://doi.org/10.1112/blms/26.5.431
15. D. M. Riley and J. F. Semple, Completion of restricted Lie algebras, Israel J. Math. 86 (1994), no. 1-3, 277–299. MR 1276139, https://doi.org/10.1007/BF02773682
16. S. Siciliano and Th. Weigel, On powerful and 𝑝-central restricted Lie algebras, Bull. Austral. Math. Soc. 75 (2007), no. 1, 27–44. MR 2309546, https://doi.org/10.1017/S000497270003896X
17. I. N. Stewart, Infinite-dimensional Lie algebras in the spirit of infinite group theory, Compositio Math. 22 (1970), 313–331. MR 0288159
18. Ian Stewart, A note on 2-step subideals of Lie algebras, Compositio Math. 27 (1973), 273–275. MR 0335584
19. 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
20. D. A. Towers, Elementary Lie algebras, J. London Math. Soc. (2) 7 (1973), 295–302. MR 0376782, https://doi.org/10.1112/jlms/s2-7.2.295
21. David Towers, Lie algebras all of whose proper subalgebras are nilpotent, Linear Algebra Appl. 32 (1980), 61–73. MR 577906, https://doi.org/10.1016/0024-3795(80)90007-5
22. David Towers, On complemented Lie algebras, J. London Math. Soc. (2) 22 (1980), no. 1, 63–65. MR 579809, https://doi.org/10.1112/jlms/s2-22.1.63
23. David A. Towers, On Lie algebras in which modular pairs of subalgebras are permutable, J. Algebra 68 (1981), no. 2, 369–377. MR 608540, https://doi.org/10.1016/0021-8693(81)90269-6