Solvable lie algebras and their subalgebra lattices

Gein, A. G.; Varea, Vicente R.

doi:10.1080/00927879208824458

Cited by 12 publications

(6 citation statements)

References 5 publications

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“…In [13], Gein and Varea showed that solvability was a subalgebra lattice property, provided that L was at least three-dimensional and the underlying field was perfect of characteristic different from 2, 3. We now have that the same is true for strong solvability.…”

Section: Corollary 22 Let F Be Perfect Then Every Solvable E-algebmentioning

confidence: 98%

Elementary Lie algebras and Lie A-algebras

Towers

Varea

2007

Journal of Algebra

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Section: Corollary 22 Let F Be Perfect Then Every Solvable E-algebmentioning

confidence: 98%

Elementary Lie algebras and Lie A-algebras

Towers

Varea

2007

Journal of Algebra

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“…So suppose that r > 1. Then, as in Lemma 3.2 of [5], there exists u 0 ∈ U such that α i (u 0 ) = α j (u 0 ) for every 1 i = j r. Pick 0 = a ∈ A, and put C = ∞ k=0 a(ad u 0 ) k , so that C is a cyclic subspace of A relative to σ(u 0 ). Now (A Ω ) αi is just the eigenspace of A Ω corresponding to the eigenvalue α i (u 0 ) relative toσ(u 0 ) for every 1 i r. Thus…”

Section: General Resultsmentioning

confidence: 99%

On Upper-Modular Subalgebras of a Lie Algebra

Bowman

Towers²,

Varea

2004

Proceedings of the Edinburgh Mathematical Society

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This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper-modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper-modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular, the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper-modular atom which is not an ideal. Finally, it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a µ-algebra.

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“…These include semi-modular, upper semi-modular, lower semi-modular, upper modular, lower modular, and their respective duals. For a selection of results on these conditions see [9,11,14,15,17,19,21,22,24,28].…”

Section: Introductionmentioning

confidence: 99%

On the subalgebra lattice of a Leibniz algebra

Siciliano

Towers

2021

Communications in Algebra

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In this paper, we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case.

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Solvable lie algebras and their subalgebra lattices

Cited by 12 publications

References 5 publications

Elementary Lie algebras and Lie A-algebras

Elementary Lie algebras and Lie A-algebras

On Upper-Modular Subalgebras of a Lie Algebra

On the subalgebra lattice of a Leibniz algebra

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