Restricted Infinitesimal Deformations of Restricted Simple Lie Algebras

Viviani, Filippo

doi:10.1142/s0219498812500910

Cited by 5 publications

(8 citation statements)

References 35 publications

(41 reference statements)

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“…Using the same notation as in [16], we let Hom Fr (V, W ) denote the set of Frobenius homomorphisms from the F-vector space V to the F-vector space W . That is…”

Section: Restricted Cochain Complexmentioning

confidence: 99%

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Restricted one-dimensional central extensions of the restricted filiform Lie algebras m0λ(p)

Evans

Fialowski

2019

Linear Algebra and its Applications

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Consider the filiform Lie algebra m 0 with nonzero Lie brackets [e 1 , e i ] = e i+1 for 1 < i < p, where the characteristic of the field F is p > 0. We show that there is a family m λ 0 (p) of restricted Lie algebra structures parameterized by elements λ ∈ F p . We explicitly describe both the ordinary and restricted 1−cohomology spaces and show that for p ≥ 3 these spaces are equal. We also describe the ordinary and restricted 2−cohomology spaces and interpret our results in the context of one-dimensional central extensions.

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“…Using the same notation as in [16], we let Hom Fr (V, W ) denote the set of Frobenius homomorphisms from the F-vector space V to the F-vector space W . That is…”

Section: Restricted Cochain Complexmentioning

confidence: 99%

“…where ϕ ∈ C 2 (g) and g, h ∈ g (see [16]). If p ≥ 3, the map H 1 * (m λ 0 (p)) → H 1 (m λ 0 (p)) is an isomorphism so that the sequence (10) decouples and the sequence…”

Section: Restricted Cohomology With λ =mentioning

confidence: 99%

Restricted one-dimensional central extensions of the restricted filiform Lie algebras m0λ(p)

Evans

Fialowski

2019

Linear Algebra and its Applications

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“…Here we use the same notation as in [12] and write Hom fr (V, W ) for the set of Frobenius homomorphisms from the F-vector space V to the F-vector space W . That is…”

Section: Restricted One-dimensional Central Extensionsmentioning

confidence: 99%

“…This table includes the Witt algebra W (1) [3,6], the Special algebras S(n) [2], Hamiltonian algebras H(n) [2], and the Contact algebras K(n) for n + 3 ≡ 0 (mod p) [6]. Finally, we remark that in [12], the map…”

Section: Case By Case Studymentioning

confidence: 99%

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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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