Graded Lie algebras with classical reductive null component

Benkart, Georgia; Gregory, Thomas B.

doi:10.1007/bf01442673

Cited by 18 publications

(186 citation statements)

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“…We note that by, for example, [9] (Lemma 6), G is transitive in its negative part. (Note that the lemmas we quote from [9] are valid for all prime characteristics.) As usual, we assume throughout that M(G) = 0 [8].…”

Section: Intermediate Resultsmentioning

confidence: 99%

On the Initial Subalgebra of a Graded Lie Algebra

Gregory¹

2014

APM

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Section: Intermediate Resultsmentioning

confidence: 99%

On the Initial Subalgebra of a Graded Lie Algebra

Gregory¹

2014

APM

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“…be as in Section 3 of [1], and let M (B(L −2 )) be as in Weisfeiler's Theorem (Theorem 1.3). We focus on…”

Section: Lemma 228 We May Assume That (Smentioning

confidence: 99%

“…(See also [3].) In [1] it is shown that for p > 5, L −1 is necessarily a restricted L ′ 0 -module. (The assertion is also true for p = 5.…”

Section: Introductionmentioning

confidence: 99%

On graded Lie algebras of characteristic three with classical reductive null component

Gregory

Кузнецов

1998

The Monster and Lie Algebras

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We consider finite-dimensional irreducible transitive graded Lie algebras L = r i=−q Li over algebraically closed fields of characteristic three. We assume that the null component L0 is classical and reductive. The adjoint representation of L on itself induces a representation of the commutator subalgebra L ′ 0 of the null component on the minus-one component L−1. We show that if the depth q of L is greater than one, then this representation must be restricted.Over algebraically closed fields F of characteristic p > 0, the classification of the finite-dimensional simple Lie algebras relies on the classification of the finite-dimensional irreducible transitive graded Lie algebras L = r i=−q L i of depth q ≧ 1 with classical reductive null component L 0 . We recall some of the progress that has been made in the classification of such Lie algebras L. In the case in which L −1 is not only irreducible but also restricted as an L 0 -module, such

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“…Using this information, we then determine L completely. There are only two isomorphism classes of simple algebras of toral rank 3 in which every 1-section with respect to every such torus is solvable, namely H(2; (2, 1); Φ(τ )) (1) and S(3; 1; Φ(τ)) (1) . We also normalize the tori of maximal toral rank in the p-envelope of these algebras ( §5).…”

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“…We also normalize the tori of maximal toral rank in the p-envelope of these algebras ( §5). Using this information, we are able to prove that the algebras in question of arbitrary toral rank are Block algebras of type L(G, 0, f) or special algebras S(m; n; Φ(τ)) (1) . The results of this note complete the proof of the result announced in [22].…”

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The classification of the simple modular Lie algebras: VI. Solving the final case

Strade

1998

Trans. Amer. Math. Soc.

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Abstract. We investigate the structure of simple Lie algebras L over an algebraically closed field of characteristic p > 7. Let T denote a torus in the p-envelope of L in Der L of maximal dimension. We classify all L for which every 1-section with respect to every such torus T is solvable. This settles the remaining case of the classification of these algebras.This paper is the last of a series of notes which are concerned with the problem of classifying the simple finite dimensional Lie algebras over an algebraically closed field of characteristic p > 7. Earlier notes have provided the preparatory material and the solution of three cases out of four. Here we will solve the remaining case. Our procedure is as follows. We define the concept of an absolute toral rank T R(K, L) of a subalgebra K of a Lie algebra L [11] and consider tori T in a p-envelope L p of a simple, not necessarily restricted Lie algebra L having maximal absolute toral rank T R(T, L p ) in L p . These tori correspond to tori of maximal dimension in the restricted simple algebras. We then consider sections L(α 1 , . . . , α t ) of L with respect to such a torus of maximal absolute toral rank. The 1-sections have been described in [11]. In [12] the possible isomorphism classes of the 2-sections have been determined. The list occurring there is exactly that of [3, (9

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Graded Lie algebras with classical reductive null component

Cited by 18 publications

References 7 publications

On the Initial Subalgebra of a Graded Lie Algebra

On the Initial Subalgebra of a Graded Lie Algebra

On graded Lie algebras of characteristic three with classical reductive null component

The classification of the simple modular Lie algebras: VI. Solving the final case

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