Some Theorems on Leibniz Algebras

Barnes, Donald W.

doi:10.1080/00927872.2010.489529

Cited by 83 publications

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“…In particular, Engel's theorem has been extended in [Ayupov and Omirov 1998;Barnes 2011;Patsourakos 2007]. In the second of these works, the classical Engel's theorem is used to give a short proof of the result for Leibniz algebras.…”

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Jacobson’s refinement of Engel’s theorem for Leibniz algebras

Bosko

Hedges

Hird

et al. 2011

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We develop Jacobson's refinement of Engel's Theorem for Leibniz algebras. We then note some consequences of the result.Since Leibniz algebras were introduced in [Loday 1993] as a noncommutative generalization of Lie algebras, one theme has been to extend Lie algebra results to Leibniz algebras. In particular, Engel's theorem has been extended in [Ayupov and Omirov 1998;Barnes 2011;Patsourakos 2007]. In the second of these works, the classical Engel's theorem is used to give a short proof of the result for Leibniz algebras. The proofs in the other two papers do not use the classical theorem and, therefore, the Lie algebra result is included in the result. In this note, we give two proofs of the generalization to Leibniz algebras of Jacobson's refinement to Engel's theorem, a short proof which uses Jacobson's theorem and a second proof which does not use it. It is interesting to note that the technique of reducing the problem to the special Lie algebra case significantly shortens the proof for the general Leibniz algebras case. This approach has been used in a number of situations [Barnes 2011]. We also note some standard consequences of this theorem. The proofs of the corollaries are exactly as in Lie algebras (see [Kaplansky 1971]). Our result can be used to directly show that the sum of nilpotent ideals is nilpotent, and hence one has a nilpotent radical. In this paper, we consider only finite dimensional algebras and modules over a field ‫.ކ‬ An algebra A is called Leibniz if it satisfies x(yz) = (x y)z + y(x z). Denote by R a and L a , respectively, right and left multiplication by a ∈ A. Then Using (1) and (2) we obtainwhere a 1 = a and a n is defined inductively as a n+1 = aa n . Furthermore, for n > 1,For any set X in an algebra, we let X denote the algebra generated by X .is a combination of terms with each term having at least 2n − 1 factors. Moreover, each of these factors is either L a or R a . Any L a to the right of the first R a can be turned into an R a using (4). Hence, any term with 2n − 1 factors can be converted into a term with either L a in the first n − 1 leading positions or R a in the last n postitions. In either case, the term is 0 and s 2n−1 = 0. Thus R a , L a is nil and hence nilpotent.Let M be an A-bimodule and let T a (m) = am and S a (m) = ma, a ∈ A, m ∈ M. The analogues of (1)- (4) hold:These operations have the same properties as L a and R a , and the associative algebra T a , S a generated by all T b , S b , b ∈ a is nilpotent if T a is nilpotent. We record this as Lemma. Let A be a finite dimensional Leibniz algebra and let a ∈ A. Let M be a finite dimensional A-bimodule such that T a is nilpotent on M. Then S a is nilpotent, and S a , T a , the algebra generated by all S b , T b , b ∈ a , is nilpotent.A subset of A which is closed under multiplication is called a Lie set.Theorem (Jacobson's refinement of Engel's theorem for Leibniz algebras). Let A be a finite dimensional Leibniz algebra and M be a finite dimensional A-bimodule. Let C be a Lie set in A such that A = C . Suppose...

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Jacobson’s refinement of Engel’s theorem for Leibniz algebras

Bosko

Hedges

Hird

et al. 2011

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“…It is interesting to note that the technique of reducing the problem to the special Lie algebra case significantly shortens the proof for the general Leibniz algebras case. This approach has been used in a number of situations [Barnes 2011]. We also note some standard consequences of this theorem.…”

Section: Conclusion and Open Questionsmentioning

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Section: Conclusion and Open Questionsmentioning

confidence: 99%

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“…For example: In [5], [14], [15], the authors study nilpotent Leibniz n-algebras, nilpotent Lie and Leibniz algebras, nilpotent n-Lie algebras, respectively; D. W. Barnes discusses Engel subalgebras of Leibniz algebras in [3], and so on. In 1996, the concept of n-Lie superalgebras was first introduced by Yu.…”

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confidence: 99%