On 𝑛-maximal subalgebras of Lie algebras

Towers, David

doi:10.1090/proc/12821

On $n$-maximal subalgebras of Lie algebras
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Proc. Amer. Math. Soc. 144 (2016), 1457-1466 Request permission

Abstract:

A $2$-maximal subalgebra of a Lie algebra $L$ is a maximal subalgebra of a maximal subalgebra of $L$. Similarly we can define $3$-maximal subalgebras, and so on. There are many interesting results concerning the question of what certain intrinsic properties of the maximal subalgebras of a Lie algebra $L$ imply about the structure of $L$ itself. Here we consider whether similar results can be obtained by imposing conditions on the $n$-maximal subalgebras of $L$, where $n>1$.

References

A. L. Agore and G. Militaru, Extending structures for Lie algebras, Monatsh. Math. 174 (2014), no. 2, 169–193. MR 3201255, DOI 10.1007/s00605-013-0537-7
Donald W. Barnes, Nilpotency of Lie algebras, Math. Z. 79 (1962), 237–238. MR 150177, DOI 10.1007/BF01193118
Donald W. Barnes, On the cohomology of soluble Lie algebras, Math. Z. 101 (1967), 343–349. MR 220784, DOI 10.1007/BF01109799
Donald W. Barnes and Humphrey M. Gastineau-Hills, On the theory of soluble Lie algebras, Math. Z. 106 (1968), 343–354. MR 232807, DOI 10.1007/BF01115083
V. A. Belonogov, Finite solvable groups with nilpotent $2$-maximal subgroups, Mat. Zametki 3 (1968), 21–32 (Russian). MR 223445
Juan C. Benjumea, Juan Núñez, and Ángel F. Tenorio, Maximal abelian dimensions in some families of nilpotent Lie algebras, Algebr. Represent. Theory 15 (2012), no. 4, 697–713. MR 2944438, DOI 10.1007/s10468-010-9260-4
Bertram Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409–434 (German). MR 64771, DOI 10.1007/BF01187387
Dmitriy V. Maksimenko, On action of outer derivations on nilpotent ideals of Lie algebras, Algebra Discrete Math. 1 (2009), 74–82. MR 2542496
Avino’am Mann, Finite groups whose $n$-maximal subgroups are subnormal, Trans. Amer. Math. Soc. 132 (1968), 395–409. MR 223446, DOI 10.1090/S0002-9947-1968-0223446-X
Eugene Schenkman, A theory of subinvariant Lie algebras, Amer. J. Math. 73 (1951), 453–474. MR 42399, DOI 10.2307/2372187
Ernest L. Stitzinger, A note on a class of Lie algebras, Math. Z. 116 (1970), 141–142. MR 272843, DOI 10.1007/BF01109957
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
D. A. Towers, A Frattini theory for algebras, Proc. London Math. Soc. (3) 27 (1973), 440–462. MR 427393, DOI 10.1112/plms/s3-27.3.440
David Towers, Lie algebras all of whose proper subalgebras are nilpotent, Linear Algebra Appl. 32 (1980), 61–73. MR 577906, DOI 10.1016/0024-3795(80)90007-5
David A. Towers, $c$-ideals of Lie algebras, Comm. Algebra 37 (2009), no. 12, 4366–4373. MR 2588856, DOI 10.1080/00927870902829023
David A. Towers, Subalgebras that cover or avoid chief factors of Lie algebras, Proc. Amer. Math. Soc. 143 (2015), no. 8, 3377–3385. MR 3348780, DOI 10.1090/S0002-9939-2015-12533-6
V. R. Varea, Lie algebras whose maximal subalgebras are modular, Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), no. 1-2, 9–13. MR 700495, DOI 10.1017/S0308210500016085
Vicente R. Varea, On modular subalgebras in Lie algebras of prime characteristic, Lie algebra and related topics (Madison, WI, 1988) Contemp. Math., vol. 110, Amer. Math. Soc., Providence, RI, 1990, pp. 289–307. MR 1079113, DOI 10.1090/conm/110/1079113
Vicente R. Varea, Lie algebras all of whose proper subalgebras are solvable, Comm. Algebra 23 (1995), no. 9, 3245–3267. MR 1335301, DOI 10.1080/00927879508825401