Sums of Simple and Nilpotent Lie Subalgebras

Bahturin, Yuri; Tvalavadze, Marina V.; Tvalavadze, T.

doi:10.1081/agb-120013332

Cited by 6 publications

(8 citation statements)

References 7 publications

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“…where the ideals g (i) are recursively defined by g (i) = [g (i−1) , g (i−1) ] for all i ≥ 1. We have g (0) = g, g (1) = [g, g], g (2) = [[g, g], [g, g]], and so on. A Lie algebra g is called nilpotent, if g k = 0 for some k. It is called solvable, if g (k) = 0 for some k. Every nilpotent Lie algebra is also solvable, and we have the inclusion…”

Section: Preliminariesmentioning

confidence: 99%

“…It seems very useful to follow the ideas in [3]. There we start with the equation dim(g) = dim g/g (1) + dim g (1) /g (2) + dim g (2) /g (3) + dim g (3) .…”

Section: Derived Length Of Nilpotent Lie Algebrasmentioning

confidence: 99%

“…After some computation we found the following Lie algebra: This Lie algebra has integer coefficients and satisfies d(g) = 4 and c(g) = 11. More precisely we have dim(g (1) ), . .…”

Section: Derived Length Of Nilpotent Lie Algebrasmentioning

confidence: 99%

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Derived length and nildecomposable Lie algebras

Burde

2012

Preprint

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We study the minimal dimension of solvable and nilpotent Lie algebras over a field of characteristic zero with given derived length k. This is motivated by questions on nildecomposable Lie algebras g = a + b, arising in the context of post-Lie algebra structures. The question is, how the derived length of g can be estimated in terms of the derived length and nilpotency classes of the two nilpotent subalgebras a and b.

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Section: Preliminariesmentioning

confidence: 99%

“…It seems very useful to follow the ideas in [3]. There we start with the equation dim(g) = dim g/g (1) + dim g (1) /g (2) + dim g (2) /g (3) + dim g (3) .…”

Section: Derived Length Of Nilpotent Lie Algebrasmentioning

confidence: 99%

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Derived length and nildecomposable Lie algebras

Burde

2012

Preprint

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“…Bahturin and Kegel [1] completely described all finite-dimensional associative algebras over a field that are sums of two simple subalgebras. Several results on sums of simple and nilpotent subalgebras of Lie algebras have been obtained in [2].…”

Section: Introductionmentioning

confidence: 99%

Semisimple decompositions of Lie algebras and prehomogeneous modules

Burde¹,

Moens²

2022

Preprint

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We study disemisimple Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra g is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous s-module for a Levi subalgebra s of g. We use the classification of prehomogeneous s-modules for simple Lie algebras s given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type A.

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“…Corresponding classes of groups were studied by Baumeister ( [1]), who showed, in particular, that any group in MA is solvable, and that a group belongs to MD if and only if it is solvable. Similar problems concerning factorisations of Lie algebras as sums of subalgebras of a certain type have been studied extensively (see, for example, [12], [13], [2], [9], [17], [19], [20], [21] and the references contained therein. )…”

Section: Introductionmentioning

confidence: 99%

Supplements to Maximal Subalgebras of Lie Algebras

Towers

2013

Communications in Algebra

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For a Lie algebra L and a subalgebra M of L we say that a subalgebra U of L is a supplement to M in L if L = M + U . We investigate those Lie algebras all of whose maximal subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first class partial results are obtained.

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Sums of Simple and Nilpotent Lie Subalgebras

Cited by 6 publications

References 7 publications

Derived length and nildecomposable Lie algebras

Derived length and nildecomposable Lie algebras

Semisimple decompositions of Lie algebras and prehomogeneous modules

Supplements to Maximal Subalgebras of Lie Algebras

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