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Lattice isomorphisms of Lie algebras

Published online by Cambridge University Press:  24 October 2008

D. A. Towers
D. A. Towers
Affiliation:
The University of Lancaster
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Extract

Let L, M be Lie algebras over the same field F and let ℒ(L), ℒ(M) denote their lattices of subalgebras. By an L-isomorphism (lattice isomorphism) of L onto M we mean an isomorphism θ: ℒ(L) → ℒ(M); that is, θ a bijective mapping from ℒ(L) onto ℒ(M) such that

and

for all subalgebras A, B of L (where A ∪ B denotes the subalgebra of L generated by A and B). We shall write A* for θ(A), the image of A ∈ ℒ(L) under an L-isomorphism from L onto M = L*. Then we may ask how closely related are L and L*. In particular, are such important concepts as semisimplicity, solvability, nilpotency and ideals preserved by L-isomorphisms?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 2 , March 1981 , pp. 285 - 292
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Barnes, D. W.Lattice isomorphisms of Lie algebras. J. Australian Math. Soc. 4 (1964), 470–475.CrossRefGoogle Scholar
(2)Barnes, D. W.On the Cohomology of soluble Lie algebras. Math. Z. 101 (1967), 343–349.CrossRefGoogle Scholar
(3)Gein, A. G.Projections of a Lie algebra of characteristic zero. Izvestija vysš. učebn. Zaved. Mat., no. 4, 191 (1978), 26–31.Google Scholar
(4)Goto, M.Lattices of subalgebras of real Lie algebras. J. Algebra 11 (1969), 6–24.CrossRefGoogle Scholar
(5)Ionescu, T.On the generators of semisimple Lie algebras. Linear Algebra and Appl. 15 (1976), 271–292.CrossRefGoogle Scholar
(6)Kolman, B.Semimodular Lie algebras. J. Sci. Hiroshima Univ. Ser. A–I 29 (1965), 149–163.Google Scholar
(7)Towers, D. A.A Frattini theory for algebras. Proc. London Math. Soc. (3) 27 (1973), 440–462.CrossRefGoogle Scholar
(8)Towers, D. A.Dimension-preserving lattice isomorphisms of Lie algebras. Communications in Algebra. (To appear).Google Scholar
(9)Towers, D. A.On complemented Lie algebras. J. London Math. Soc. 22 (1980), 63–65.CrossRefGoogle Scholar