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Lattice isomorphisms of Malcev algebras

Published online by Cambridge University Press:  14 November 2011

Alberto Elduque
Alberto Elduque
Affiliation:
Departamento de Matematica Aplicada, E.T.S.I.I., Universidad de Zaragoza, 50015 Zaragoza, Spain
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Synopsis

The lattice of subalgebras of a Malcev algebra determines to a great extent the structureof the algebra. It is shown that conditions such as nilpotency, solvability or semisimplicity are almost characterised by means of conditions on this lattice. This enables us to study the relationship between Malcev algebras with isomorphic lattices of subalgebras.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 37 - 50
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Amayo, R. K.. Quasiideals of Lie algebras, I and II. Proc. London Math. Soc. (3) 33 (1976), 28–64.CrossRefGoogle Scholar
2Amayo, R. K. and Schwarz, J.. Modularity in Lie algebras. Hiroshima Math. J. 10 (1980), 311–322.Google Scholar
3Barnes, D. W.. Lattice isomorphisms of Lie algebras. J. Austral. Math. Soc. 4 (1964), 470–475.CrossRefGoogle Scholar
4Carlsson, R.. Malcev-moduln. J. Reine Angew. Math. 286 (1976), 199–210.Google Scholar
5Elduque, A.. On Malcev algebras in which all subideals are ideals. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 353–363.CrossRefGoogle Scholar
6Elduque, A.. On supersolvable Malcev algebras. Comm. Algebra 14 (1986), 311–321.CrossRefGoogle Scholar
7Elduque, A.. On maximal subalgebras of central simple Malcev algebras. J. Algebra 99 (1986), 80–88.Google Scholar
8Malek, A. A. El. On the Frattini subalgebra of a Malcev algebra. Arch. Math. Basel 37 (1981), 306–315.Google Scholar
9Malek, A. A. El. Modularity in Malcev algebras. Arch. Math. Basel 44 (1985), 233–242.CrossRefGoogle Scholar
10Farnsteiner, R.. On ad-semisimple Lie algebras. J. Algebra 83 (1983), 510–519.Google Scholar
11Gejn, A. G.. Projections of a Lie algebra of characteristic zero. Izv. Vyssh. Uchebn. Zaved. Mat. 191 (1978), 26–31.Google Scholar
12Gejn, A. G.. Modular subalgebras and projections of locally finite-dimensional Lie algebras over fields of characteristic zero. (Russian) Ural. Gos. Univ. Mat. Zap. 13 (1983), 39–51.Google Scholar
13Goto, M.. Lattices of subalgebras of real Lie algebras. J. Algebra 11 (1969), 6–24.CrossRefGoogle Scholar
14Gratäer, G.. General Lattice Theory (Stuttgart: Birkhauser 1978).Google Scholar
15Ionescu, T.. On the generators of semisimple Lie algebras. Linear Algebra Appl. 15 (1976), 271–292.CrossRefGoogle Scholar
16Jacobson, N.. Lie algebras (New York: Interscience, 1962).Google Scholar
17Kuranishi, M.. Two elements generations on semisimple Lie groups. Kodai Math. Sem. Rep. 5–6 (1949), 9–10.Google Scholar
18Kuzmin, E. N.. Mal'tsev algebras and their representations. Algebra i Logika 1 (1968), 48–69.Google Scholar
19Kuzmin, E. N.. Simple Mal'tsev algebras over a field of characteristic zero. Dokl. Akad. Nauk. UzSSR 181 (1968), 1324–1326.Google Scholar
20Lashi, A. A.. On Lie algebras with modular lattices of subalgebras. J. Algebra 99 (1986), 80–88.CrossRefGoogle Scholar
21Sagle, A. A.. Malcev algebras. Trans. Amer. Math. Soc. 101 (1961), 426–458.CrossRefGoogle Scholar
22Towers, D. A.. Lattice isomorphisms of Lie algebras. Math. Proc. Cambridge Philos. Soc. 89 (1981), 283–292; (Corrigendum: 95 (1984), 511).Google Scholar
23Towers, D. A.. Almost nilpotent Lie algebras. Glasgow Math. J. 29 (1987), 7–11.CrossRefGoogle Scholar