We show that for fields that are of characteristic 0 or algebraically closed of characteristic greater than 5, that certain classes of Leibniz algebras are 2-recognizeable. These classes are solvable, strongly solvable and supersolvable. These same results hold in Lie algebras and in general for groups.Key Words: 2-recognizeable, strongly solvable, supersolvable, Leibniz algebras I. PRELIMINARIES A property of algebras is called n-recognizeable if whenever all the n generated subalgebras of algebra L have the property, then L also has the property. An analogous definition holds for classes of groups. In Lie algebras, nilpotency is 2-recognizeable due to Engel's theorem and the same holds for Leibniz algebras. For Lie algebras, solvability, strong solvability and supersolvability are 2-recognizeable when they are taken over a field of characteristic 0 or an algebraically closed field of characteristic greater than 5. These results are shown in [7] and [12] using different methods. The purpose of this work is to extend these results to Leibniz algebras. Corresponding results in group theory are shown in [8] and [9].The definition of Leibniz algebra can be given in terms of the left multiplications being derivations. A theme in this work is that assumptions will be given in terms of the left multiplications. 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.
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 by Loday in [6] as a noncommutative generalization of Lie algebras, one theme is to extend Lie algebra results to Leibniz algebras. In particular, Engel's theorem has been extended in [1], [3], and [7]. In [3], the classical Engel's theorem is used to give a short proof of the result for Leibniz algebras. The proofs in [1] and [7] 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, see [2]. We also note some standard consequences of this theorem. The proofs of the corollaries are exactly as in Lie algebras (see [5]). 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 F.An algebra A is called Leibniz if it satisfies x(yz) = (xy)z + y(xz). Denote by R a and L a , respectively, right and left multiplication by
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Ideals that share properties with the Frattini ideal of a Leibniz algebra are studied. Similar investigations have been considered in group theory. However most of the results are new for Lie algebras. Many of the results involve nilpotency of these algebras.
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