On the Structure of the Graded Lie Algebra Associated with a Noncontractible Filtration

Skryabin, Serge

doi:10.1006/jabr.1997.7124

Cited by 8 publications

(5 citation statements)

References 14 publications

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“…Now [21,Corollary 6.34] (which generalizes earlier work of Kuznetsov [3], Weisfeiler [24], and Skryabin [11]) says that L is one of sl (2), W (1; n), H (2; n; Φ) (2) for some n and Φ. As TR(L) = 3 we have L sl (2).…”

Section: Case (A): Lie Algebras Without Nonsolvable 1-sectionsmentioning

confidence: 62%

Simple Lie algebras of small characteristic IV.

Premet

Strade

2004

Journal of Algebra

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Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic p > 3 and T a torus of maximal dimension in the p-envelope of L in Der L. In this paper we describe the T -semisimple quotients of the 2-sections of L relative to T and prove that if all 1-sections of L relative to T are compositionally classical or solvable then L is either classical or a Block algebra or a filtered Lie algebra of type S.

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Section: Case (A): Lie Algebras Without Nonsolvable 1-sectionsmentioning

confidence: 62%

Simple Lie algebras of small characteristic IV.

Premet

Strade

2004

Journal of Algebra

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“…To show that j = 1, we will, for a contradiction, assume that j = 2. We begin by using an inductive argument from [13,Lemma 2.14] to show that the centralizer of S s in L <0 is zero. Denote by Z the centralizer of S s in L. Then Z = ⊕Z i is a homogeneous subspace of L. Since S s is stable under ad L ≧0 , Z is, as well.…”

Section: Conclusion Of the Proof Of The Main Theoremmentioning

confidence: 99%

“…(See Corollary 1.7.) But then we would have [L 0 , S s ] = [[S 2 , Z −2 ], S s ] = [S 2 , [Z −2 , S s ]] = 0, to contradict[13, Lemma 2.13]. (See also Lemma 2.28.)…”

mentioning

confidence: 96%

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

Gregory

Кузнецов

2019

Journal of Algebra

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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.

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“…3] for producing quotients of subalgebras in a graded Lie algebra g = r i=−q g i . It has been applied subsequently in several other settings; for example, it was used by Skryabin in [Sk2] to derive information about the solvable radical of g 0 .…”

Section: The Weisfeiler Radicalmentioning

confidence: 99%

The recognition theorem for graded Lie algebras in prime characteristic

Benkart¹,

Gregory²,

Premet³

2009

Memoirs of the AMS

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Chapter 3. The Contragredient Case 3.1. Introduction 3.2. Results on modules for three-dimensional Lie algebras 3.3. Primitive vectors in g 1 and g −1 3.4. Subalgebras with a balanced grading 3.5. Algebras with an unbalanced grading Chapter 4. The Noncontragredient Case 4.1. General assumptions and notation 4.2. Brackets of weight vectors in opposite gradation spaces 4.3. Determining g 0 and its representation on g −1 4.4. Additional assumptions 4.5. Computing weights of b − -primitive vectors in g 1 4.6. Determination of the local Lie algebra 4.7. The irreducibility of g 1 4.8. Determining the negative part when g 1 is irreducible 4.9. Determining the negative part when g 1 is reducible 4.10. The case that g 0 is abelian 4.11. Completion of the proof of the Main Theorem Bibliography

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On the Structure of the Graded Lie Algebra Associated with a Noncontractible Filtration

Cited by 8 publications

References 14 publications

Simple Lie algebras of small characteristic IV.

Simple Lie algebras of small characteristic IV.

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

The recognition theorem for graded Lie algebras in prime characteristic

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