On Cartan subalgebras of Lie algebras

Barnes, Donald W.

doi:10.1007/bf01109800

Cited by 57 publications

(37 citation statements)

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“…Let S P (x) denote the largest subspace of P on which adx is split. It is well-known that S P (x) is a subalgebra of P and that there exists a subspace K P (x) of P invariant under ad x and such that L = S P (x) ® K P (x), see [4]. It is easy to see that …”

Section: Lemma 55 Let L Be a Lie Algebra And Let N < L Be Such Thatmentioning

confidence: 99%

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Lower semimodular Lie algebras

Varea¹

1999

Proceedings of the Edinburgh Mathematical Society

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This paper is concerned with the relationship between the properties of the subalgebra lattice C(L) of a Lie algebra L and the structure of L. If the lattice C(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice C(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A < L and B is maximal in A (Lie algebras satisfying this condition are called sj;-algebras). Our aim is to characterize lower semimodular Lie algebras and s^-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modular maximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.

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Section: Lemma 55 Let L Be a Lie Algebra And Let N < L Be Such Thatmentioning

confidence: 99%

“…( By Towers [16], we have that L/(j>(L) is supersolvable. Then, by [4], it follows that L is supersolvable. (ii): Let \ff < F and let L e s*.…”

Section: S^-algebrasmentioning

confidence: 99%

Lower semimodular Lie algebras

Varea¹

1999

Proceedings of the Edinburgh Mathematical Society

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“…Indeed, if Fis algebraically closed, by Theorem 5.3 we have that H is a Cartan subalgebra of A if and only if//contains a set {e1;..., et} or pairwise orthogonal primitive idempotents whose sum is 1, and //=2 Au where 2 A¡ is the Pierce decomposition of A relative to the e¡. However, to construct Cartan subalgebras in a more general situation, we will follow the work of Barnes in the Lie case [1].…”

Section: (55) a Subalgebra H Of A Is A Cartan Subalgebra If (I) H Ismentioning

confidence: 99%

“…However, in the Lie case as in the Jordan case, we are restricted to ground fields of characteristic not 2. Finally, following Barnes [1], we will see that we can characterize our Cartan subalgebras as "minimal Engel subalgebras".…”

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confidence: 99%

On Cartan Subalgebras of Alternative Algebras

Foster¹

1971

Transactions of the American Mathematical Society

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Abstract. In 1966, Jacobson introduced the notion of a Cartan subalgebra for finite-dimensional Jordan algebras with unity over fields of characteristic not 2. Since finite-dimensional Jordan, alternative, and Lie algebras are known to be related through their structure theories, it would seem logical that such an analogue would also exist for finite-dimensional alternative algebras. In this paper, we show that this is the case. Moreover, the linear transformation we define that plays the role in alternative algebras that "ad ( )" plays in Lie algebras is identical with that used in the Jordan theory, and can be used in the Lie case as well. Hence we define Cartan subalgebras relative to this linear transformation for finite-dimensional alternative, Jordan, and Lie algebras, and observe that in the Lie case, they coincide with the classical definition of a Cartan subalgebra.

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“…Since a Lie algebra is nilpotent if and only if all its maximal subalgebras are ideals [1], M(L) consists precisely of those subalgebras of L which are nilpotent and satisfy (1). The minimal members of M(L) are a Lie algebra counterpart to system normalizers of finite solvable groups.…”

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confidence: 99%