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Lower semimodular Lie algebras

Published online by Cambridge University Press:  20 January 2009

V. R. Varea
V. R. Varea
Affiliation:
Department of Mathematics, University of Zaragoza, 50009 Zaragoza, Spain, E-mail address: varea@posta.unizar.es
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This paper is concerned with the relationship between the properties of the subalgebra lattice ℒ(L) of a Lie algebra L and the structure of L. If the lattice ℒ(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice ℒ(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A ≤ L and B is maximal in A (Lie algebras satisfying this condition are called sχ-algebras). Our aim is to characterize lower semimodular Lie algebras and sχ-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modularmaximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 521 - 540
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Amayo, R. K., Quasi-ideals of Lie algebras I, Proc. London Math. Soc. 33 (1976), 28–36.CrossRefGoogle Scholar
2.Amayo, R. K., Quasi-ideals of Lie algebras II, Proc. London Math. Soc. 33 (1976), 37–64.CrossRefGoogle Scholar
3.Amayo, R. K. and Schwarz, J., Modularity in Lie algebras, Hiroshima Math. J. 10 (1980), 311–322.Google Scholar
4.Barnes, D., On Cartan subalgebras of Lie algebras, Math. Z. 101 (1967), 350–355.Google Scholar
5.Barnes, D. and Newell, M. L., Some theorems on saturated homomorphs of soluble Lie algebras, Math. Z. 115 (1970), 179–187.Google Scholar
6.Benkart, G., Isaacs, I. M. and Osborn, J. M., Lie algebras with self-centralizing ad-nilpotent elements, J. Algebra 57 (1979), 279–309.CrossRefGoogle Scholar
7.Benkart, G. and Osborn, J. M., Rank One Lie algebras, Ann. of Math. (2) 119 (1984), 437–463.Google Scholar
8.Farnsteiner, R., Ad-semisimple Lie algebras, J. Algebra 83 (1983), 510–519.Google Scholar
9.Gein, A. G., Semimodular Lie algebras, Sibirsk. Mat. Zh. 17 (1976), 243–248; (translated in Siberian Math. J. 17 (1976), 189–193).Google Scholar
10.Gein, A. G., Modular rule and relative complements in the lattice of a Lie algebra, Izv. Vyssh. Uchebn Zaved. Mat. 383 (1987), 18–25; (translated in Soviet Math. 331 (1987), 22–32).Google Scholar
11.Gein, A. G., On modular subalgebras of Lie algebras, Ural Gos. Univ. Mat. Zap. 14 (1987), 27–33.Google Scholar
12.Jacobson, N., Lie algebras (New York: Wiley, Interscience 1962).Google Scholar
13.Strade, H. and Farnsteiner, R., Modular Lie algebras and their Representations. (Textbooks and Monographs, vol. 116, New York: Dekker 1988).Google Scholar
14.Towers, D. A., A Frattini theory for algebras, Proc. London Math. Soc. 27 (1973), 440–462.Google Scholar
15.Towers, D. A., Lie algebras all of whose maximal subalgebras have codimension one., Proc. Edinburgh Math. Soc. 24 (1981), 217–219.Google Scholar
16.Towers, D. A., On ideally finite Lie algebras which are lower semimodular, Proc. Edinburgh Math. Soc. (2) 28 (1985), 9–11.CrossRefGoogle Scholar
17.Varea, V. R., Lie algebras whose maximal subalgebras are modular, Proc. Roy. Soc. Edinburgh Sect. A94 (1983), 9–13.CrossRefGoogle Scholar
18.Varea, V. R., Lie algebras none of whose Engel subalgebras are in intermediate position, Comm. in Algebra 15(12) (1987), 2529–2543.Google Scholar
19.Varea, V. R., The subalgebra lattice of a supersolvable Lie algebra, in Lie algebras (Madison 1987, Springer Lectures Notes in Math. 1373, 1989), 81–92.Google Scholar
20.Varea, V. R., On modular subalgebras in Lie algebras of prime characteristic, Contemp. Math. 110 (1990), 289–307.Google Scholar
21.Varea, V. R., Modular subalgebras, quasi-ideals and inner ideals in Lie algebras of prime characteristic, Comm. in Algebra 21(11) (1993), 4195–4218.CrossRefGoogle Scholar
22.Varea, V. R., Lie algebras whose proper subalgebras are either semisimple, abelian or almost-abelian, Hiroshima Math. J. 24 (1994), 221–241.Google Scholar
23.Varea, V. R., Lie algebras all of whose proper subalgebras are solvable, Comm. Algebra 23(9) (1995), 3245–3267.CrossRefGoogle Scholar
24.Varea, V. R., Supersimple and upper semimodular Lie algebras, Comm. Algebra 23(6) (1995), 2323–2330.Google Scholar
25.Winter, D., Abstract Lie Algebras (MIT Press, Cambridge, Mass., 1972).Google Scholar
26.Winter, D. J., Cartan Decompositions and Engel subalgebra Triangulability, J. Algebra 62 (1990), 400–417.Google Scholar