Deformations of the Restricted Melikian Lie Algebra

Viviani, Filippo

doi:10.1080/00927870802466777

Cited by 7 publications

(10 citation statements)

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“…The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [33] (building upon [30], [31] and [32]) the infinitesimal deformations of the simple finite group schemes associated to simple and restricted Lie algebras of Cartan type.…”

Section: Deformations Of Lie Algebras Of Cartan Typementioning

confidence: 99%

Simple finite group schemes and their infinitesimal deformations

Viviani

2008

Preprint

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We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.

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Section: Deformations Of Lie Algebras Of Cartan Typementioning

confidence: 99%

Simple finite group schemes and their infinitesimal deformations

Viviani

2008

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“…The only restrictable algebra in this family is the algebra M = M(1, 1), and M has no non-trivial one-dimensional central extensions as a Lie algebra [9]. We refer the reader to [11] for an explicit description of this algebra.…”

Section: Introductionmentioning

confidence: 99%

Restricted one-dimensional central extensions of restricted simple Lie algebras

Evans

Fialowski

2017

Linear Algebra and its Applications

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The one-dimensional central extensions of a Lie algebra g defined over a field F are classified by the Lie algebra cohomology group H 2 (g) = H 2 (g, F) where F is taken as a trivial g-module. The restricted one-dimensional central extensions of a restricted Lie algebra g over F are likewise classified by the restricted Lie algebra cohomology group H 2 * (g) = H 2 * (g, F). We refer the reader to [3] and [4] for descriptions of the complexes used to compute the ordinary and restricted Lie algebra cohomology groups as well as a review of the correspondence between one-dimensional central extensions (restricted central extensions) and the second cohomology (restricted cohomology) group. In particular, we adopt the notation and terminology in [3].The dimensions of the ordinary cohomology groups H 2 (g) for finite dimensional simple restricted Lie algebras are known [1,2,5,6,9]. Following the technique used in [12], we use these results along with Hochschild's six term exact sequence relating the first two ordinary and restricted cohomology groups to analyze the restricted cohomology group H 2 * (g). Our theorem states that if g is a restricted simple Lie algebra, this sequence reduces to a short exact sequence relating H 2 (g) and H 2 * (g). In the case that H 2 (g) = 0, we explicitly describe the cocycles spanning H 2 * (g). If H 2 (g) = 0, we give a procedure for describing a basis for H 2 * (g). The paper is organized as follows. Section 2 gives an overview of the classification of finite dimensional simple restricted Lie algebras g defined over fields of characteristic p ≥ 5. Section 3 contains the statement and proof of the theorem relating H 2 (g) and H 2 * (g) as well as an explicit description of the cocycles spanning H 2 * (g) when H 2 (g) = 0. Section 4 outlines a procedure for describing a basis for H 2 * (g) in the case where H 2 (g) = 0. Restricted Simple Lie AlgebrasFinite dimensional simple Lie algebras over fields of characteristic zero were classified more than a century ago in the work of Killing and Cartan. The well known classification theorem states that in characteristic zero, any simple Lie algebra is isomorphic to one of the linear Lie algebras A l , B l , C l or D l (l ≥ 1),

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“…We prove these theorems using the computation of the second ordinary cohomology group H 2 (g, g) that we performed in [VIV1], [VIV2] and [VIV3], together with the 6-term exact sequence of Hochschild (see section 2.1) that relates the ordinary and restricted cohomology.…”

Section: Introductionmentioning

confidence: 99%