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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

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