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 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

Automorphisms and Derivations of Leibniz Algebras

“…An advance of Theorems 3.8 and 3.10 was presented in [14]. We also have found in the literature the reference [2], where some results here proved are showed with another techniques.…”

“…Proof. By Lemma 3.11 it follows that J + D(J) is an ideal of Leibniz algebra L. We have (J + D(J)) (2) = [J + D(J), J + D(J)] ⊆ J + [D(J), D(J)] ⊆ J + D 2 (J (2) ) .…”

A Frattini Theory for Leibniz Algebras

“…is a Lie set whose span is A 2 and left multiplication by each s in the Lie set is nilpotent on A 2 . Hence A 2 is nilpotent by the theorem in [6], which is a contradiction. Hence B is the unique minimal ideal in A.…”

“…is a Lie set, A 2 is nilpotent by [6], a contradiction. Hence t = 1 and A = B + C = B + (M 1 + D) where B is a minimal ideal in A, M 1 is a minimal ideal in C, and D is a one dimensional subalgebra with basis x.…”

“…The sum of nilpotent ideals is nilpotent by Corollary 3 of [6]. Therefore, define the nilradical of A to be the largest nilpotent ideal of A and denote it by Nil(A).…”