On the cohomology of restricted lie algebras

Feldvoss, Jörg

doi:10.1080/00927879108824299

Cited by 24 publications

(13 citation statements)

References 34 publications

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“…(see [8,Proposition 2.7]). In particular, since span m λ 2 (p) [p] ⊆ Fe p ⊆ m λ 2 (p), m λ 2 (p) , the ordinary and the restricted 1-cohomology spaces coincide.…”

Section: Restricted Cochain Complexmentioning

confidence: 99%

Cohomology of Restricted Filiform Lie Algebras m₂^λ(p)

Evans¹,

Fialowski²

2019

SIGMA

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For the p-dimensional filiform Lie algebra m 2 (p) over a field F of prime characteristic p ≥ 5 with nonzero Lie brackets [e 1 , e i ] = e i+1 for 1 < i < p and [e 2 , e i ] = e i+2 for 2 < i < p − 1, we show that there is a family m λ 2 (p) of restricted Lie algebra structures parameterized by elements λ ∈ F p . We explicitly describe bases for the ordinary and restricted 1-and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and [p]-operations in the corresponding restricted one-dimensional central extensions.

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“…(see [8,Proposition 2.7]). In particular, since span m λ 2 (p) [p] ⊆ Fe p ⊆ m λ 2 (p), m λ 2 (p) , the ordinary and the restricted 1-cohomology spaces coincide.…”

Section: Restricted Cochain Complexmentioning

confidence: 99%

Cohomology of Restricted Filiform Lie Algebras m₂^λ(p)

Evans¹,

Fialowski²

2019

SIGMA

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“…Denote the one-dimensional center of L by Z. If L is 2-unipotent, then the claim follows from [2,Corollary 5.2]. Suppose now that L is not 2-unipotent.…”

Section: Proposition 5 Every Nilpotent Restricted Derivation Of a Thmentioning

confidence: 96%

“…Let F be any field of characteristic 2 and let L be the three-dimensional Heisenberg algebra h 1 (F) = Fx + Fy + Fz defined by [x, y] = z ∈ Z(L). Since L ′ is central, we have that (2) L [2] ⊆ Z(L) = Fz . Now, observe that D 2 = αD and therefore D n = α n−1 D for every positive integer n. As D is nilpotent, we conclude that α = 0.…”

Section: Proposition 5 Every Nilpotent Restricted Derivation Of a Thmentioning

confidence: 99%

“…In fact, Tôgô's result is more general (see [11,Theorem 1]) and is a refinement of a theorem of E. Schenkman that establishes the existence of outer derivations for non-zero finite-dimensional nilpotent Lie algebras (see [7,Theorem 4]). Much later the first author proved a restricted analogue of Schenkman's result for p-unipotent restricted Lie algebras (see [2,Corollary 5.2]). (Here we follow [1,Section I.4, Exercise 23, p. 97] by calling a restricted Lie algebra (L, [p]) p-unipotent if for every x ∈ L there exists some positive integer n such that x [p] n = 0.)…”

Section: Introductionmentioning

confidence: 99%

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Outer restricted derivations of nilpotent restricted Lie algebras

Feldvoss

Siciliano

Weigel

2012

Proc. Amer. Math. Soc.

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Abstract. In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Tôgô on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschütz on the existence of p-power automorphisms of p-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.

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“…For the first two isomorphisms we are using [Fe,(2.7) Proposition] and the fact that p-powers of elements in n + are zero. The last isomorphism follows because H 1 (n + , k) is spanned by the classes of e * 1 and e * 2 , with weights −1 and −2.…”

Section: Extensions Involving L(0) As In the Proof Of Proposition 2mentioning

confidence: 99%