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

Abstract

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...

On some structures of Leibniz algebras

“…The second method greatly shortens the proof. The short proof given here shows the result using Lie sets and follows that in [12]. A Lie subset of a Leibniz algebra A is a subset that is closed under multiplication.…”

“…As an immediate consequence of Proposition 4.2, we have Several authors have given various forms of Engel's theorem for Leibniz algebras. The Leibniz versions were proven without using the standard Lie algebra result in [4] and [27], and using the standard result in [8] and [12]. The second method greatly shortens the proof.…”

2-recognizeable classes of Leibniz algebras

“…Thus, that nilpotency is 2-recognizeable in Leibniz algebras follows from all left multiplications being nilpotent, Engel's theorem. This result, shown in several places, can be cast as in Jacobson's refinement to Engel's theorem for Lie algebras, see [6], a result that we use.…”

Conjugacy of Cartan Subalgebras in Solvable Leibniz Algebras and Real Leibniz Algebras

“…Proof If L is nilpotent of length less than three, then the result holds by the previous theorem, so let L be nilpotent of As D in nilpotent, D ⊆ K, we show K is a Cartan subalgebra of L. Assume y ∈ N L (K), but y / ∈ K. Then xy ∈ K, but y / ∈ S x , so xy / ∈ S x , contradiction. So y ∈ K thus K is its own normalizer in L. K = K ∩ (D + N (L)) = D + (K ∩ N (L)), and since D is nilpotent by definition, and (K ∩ N (L)) is a nilpotent ideal of K, K is nilpotent by [4]. Now assume D is contained in two Cartan subalgebras of L, E 1 and E 2 .…”