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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -generator subalgebra is an -step subideal, J. Algebra 31 (1974), 517-542. MR 0347919 (50:419)
2. R.K. Amayo and I. Stewart, Infinite-dimensional Lie algebras, Noordhoff International Publishing, Leyden, 1974. MR 0396708 (53:570)
3. R. Baer, Situation der Untergruppen und Struktur der Gruppe, S. B. Heidelberg. Akad. Wiss. 2 (1933), 12-17.
4. Yu. Bahturin and A. Olshanskii, Large restricted Lie algebras, J. Algebra 310 (2007), 413-427. MR 2307801 (2008b:17027)
5. Y. Barnea and I.M. Isaacs, Lie algebras with few centralizer dimensions, J. Algebra 259 (2003), 284-299. MR 1953720 (2004a:17005)
6. G. Benkart and I.M. Isaacs, Lie algebras with nilpotent centralizers, Can. J. Math. 31 (1979), 929-941. MR 546949 (81c:17022)
7. K. Bowman, D.A. Towers, and V.R. Varea, On flags and maximal chains of lower modular subalgebras of Lie algebras, J. Lie Theory 17 (2007), 605-616. MR 2352003 (2008h:17023)
8. R. Dedekind, Über Gruppen deren sämtliche Teiler Normalteiler sind, Math. Ann. 48 (1897), 548-561. MR 1510943
9. M. Honda, On Lie algebras in which every subalgebra is a subideal, Hiroshima Math. J. 18 (1988), 655-660. MR 991251 (90f:17029)
10. M. Lincoln and D.A. Towers, Frattini theory for restricted Lie algebras, Arch. Math. (Basel) 45 (1985), 451-457. MR 815984 (88d:17007)
11. S. Liu, On algebras in which every subalgebra is an ideal, Acta. Math. Sinica 14 (1964), 532-537. MR 0172899 (30:3115)
12. D.L. Outcalt, Power-associative algebras in which every subalgebra is an ideal, Pac. J. Math. 20 (1967), 481-485. MR 0207784 (34:7599)
13. D.M. Riley, Restricted Lie algebras of maximal class, Bull. Austral. Math. Soc. 59 (1999), 217-223. MR 1680807 (2000d:17023)
14. D.M. Riley and J.F. Semple, The coclass of a restricted Lie algebra, Bull. London Math. Soc. 26 (1994), 431-437. MR 1308359 (96c:17014)
15. -, Completion of restricted Lie algebras, Israel J. Math. 86 (1994), 277-299. MR 1276139 (95d:17019)
16. S. Siciliano and Th. Weigel, On powerful and -central restricted Lie algebras, Bull. Austral. Math. Soc. 75 (2007), 27-44. MR 2309546 (2008b:17030)
17. I. Stewart, Infinite-dimensional Lie algebras in the spirit of infinite group theory, Compositio Mathematica 22 (1970), 313-331. MR 0288159 (44:5357)
18. -, A note on -step subideals of Lie algebras, Compositio Mathematica 27 (1973), 273-275. MR 0335584 (49:365)
19. H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Marcel Dekker, New York, 1988. MR 929682 (89h:17021)
20. D.A. Towers, Elementary Lie algebras, J. London Math. Soc. (2) 7 (1973), 295-302. MR 0376782 (51:12957)
21. -, Lie algebras all of whose proper subalgebras are nilpotent, Linear Algebra Appl. 32 (1980), 61-73. MR 577906 (81f:17003)
22. -, On complemented Lie algebras, J. London Math. Soc. (2) 22 (1980), 63-65. MR 579809 (81h:17006)
23. -, On Lie algebras in which modular pairs of subalgebras are permutable, J. Algebra 68 (1981) 369-377. MR 608540 (82e:17010)