Lie metabelian restricted universal enveloping algebras

Siciliano, Salvatore; Spinelli, Ernesto

doi:10.1007/s00013-005-1315-0

Cited by 12 publications

(30 citation statements)

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“…Moreover, by Theorem 1 of [4] it is clear that 2) implies 1). Now, suppose dl Lie (u(L)) = log 2 ( p + 1) .…”

Section: Lemma 4 Let L Be a Restricted Lie Algebra Over A Field F Of mentioning

confidence: 92%

“…In general, the computation of the Lie derived length of u(L) is a hard task: some results on this subject can be found in [2,[4][5][6]. In particular, in [6] it was established when u(L) is Lie metabelian and, in [5], when u(L) is Lie centrally metabelian (provided p > 2).…”

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confidence: 97%

“…In particular, in [6] it was established when u(L) is Lie metabelian and, in [5], when u(L) is Lie centrally metabelian (provided p > 2). Moreover, the second author proved in [4] that, if L is not abelian, then…”

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confidence: 98%

“…Subsequently, it has been proved in [4] that u(L) is strongly Lie solvable if and only if the commutator subalgebra L of L is finite-dimensional and p-nilpotent (that is,…”

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confidence: 99%

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Restricted Enveloping Algebras with Minimal Lie Derived Length

Catino

Siciliano

Spinelli

2009

Algebr Represent Theor

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Let L be a non-abelian restricted Lie algebra over a field of characteristic p > 0 and let u(L) denote its restricted enveloping algebra. In Siciliano (Publ Math (Debr) 68: [503][504][505][506][507][508][509][510][511][512][513] 2006) it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least log 2 ( p + 1) . In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound.

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“…Moreover, by Theorem 1 of [4] it is clear that 2) implies 1). Now, suppose dl Lie (u(L)) = log 2 ( p + 1) .…”

Section: Lemma 4 Let L Be a Restricted Lie Algebra Over A Field F Of mentioning

confidence: 92%

mentioning

confidence: 97%

mentioning

confidence: 98%

“…Subsequently, it has been proved in [4] that u(L) is strongly Lie solvable if and only if the commutator subalgebra L of L is finite-dimensional and p-nilpotent (that is,…”

mentioning

confidence: 99%

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Restricted Enveloping Algebras with Minimal Lie Derived Length

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Siciliano

Spinelli

2009

Algebr Represent Theor

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“…This is certainly the case whenever A and B are group algebras; to see this, it is enough to note that a Lie metabelian group algebra is Lie nilpotent of class at most 3 (see [2,Lemma 1]). Moreover, the answer to the above question is positive also when A and B are Lie metabelian restricted universal envelopping algebras, even if such algebras are not necessarily Lie nilpotent (see the example given in [6]). Obviously if A is a Lie metabelian algebra such that γ 2 (A)γ 3 (A) = 0 then dl L (A ⊗ A) = 4.…”

Section: Vol 89 (2007)mentioning

confidence: 99%

A note on the tensor product of Lie soluble algebras

2007

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D. M. Riley proved in [3] that, if A and B are either Lie nilpotent or Lie metabelian algebras, then their tensor product A ⊗ B is Lie soluble and obtained bounds on the Lie derived length of A ⊗ B. The aim of the present note is to improve Riley's bounds; moreover we consider also the cases in which A and B are either strongly Lie soluble or strongly Lie nilpotent algebras. Mathematics Subject Classification (2000). 16R40.Keywords. Tensor product, Lie soluble algebras, Lie nilpotent algebras.1. Introduction. Throughout this note, by algebra we mean an associative algebra over a field F . Let R be an algebra. We consider the Lie algebra associated with R by setting [x, y] = xy − yx for all x, y ∈ R. As usual, for all subspaces V and W of R we denote by [V, W ] the subspace of R generated by the set of Lie commutators [x, y], with x ∈ V and y ∈ W . Moreover, we denote by V W the subspace of R generated by the set of products xy, where x ∈ V and y ∈ W . Put δ 0 (R) = R and define by induction δ n+1 (R) = [δ n (R), δ n (R)]. We say that R is Lie soluble if there exists an integer m such that δ m (R) = 0. The smallest integer with this property is called the Lie derived length of R and is denoted by dl L (R). In particular, R is said to be Lie metabelian if dl L (R) ≤ 2. Recall also that the Lie lower central chain of R is defined by γ 1 (R) = R and γ n+1 (R) = [γ n (R), R] and R is Lie nilpotent of class c if γ c+1 (R) = 0 and γ c (R) = 0. The integer c is called the Lie nilpotency class of R and is usually denoted by cl L (R).

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On Lie solvable restricted enveloping algebras

Siciliano¹

2007

Journal of Algebra

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In this note we study the Lie derived lengths of a restricted enveloping algebra u(L), for a non-abelian restricted Lie algebra L over a field of positive characteristic p. For p > 2 we show that if the Lie derived length of u(L) is minimal then u(L) is Lie nilpotent. Moreover, we investigate the case when the strong Lie derived length of u(L) is minimal. For odd p we establish a classification of Lie centrally metabelian restricted enveloping algebras.

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Lie metabelian restricted universal enveloping algebras

Abstract: We characterize the restricted Lie algebras L whose restricted universal enveloping algebra u(L) is Lie metabelian. Moreover, we show that the last condition is equivalent to u(L) being strongly Lie metabelian

Cited by 12 publications

References 5 publications

Restricted Enveloping Algebras with Minimal Lie Derived Length

Restricted Enveloping Algebras with Minimal Lie Derived Length

A note on the tensor product of Lie soluble algebras

On Lie solvable restricted enveloping algebras

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