The Lie algebra analogue to the Schur multiplier has been investigated in a number of recent articles. We consider the multipliers of Lie algebras of maximal class, classifying these algebras with a certain additional property. The classification leads to a conjecture about a bound on the dimension of the multiplier for each of these algebras and also for p-groups of maximal class. The conjectures are then shown to hold.

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...

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

We consider the Schur multipliers of finite dimensional nilpotent Lie algebras. If the algebra has dimension greater than one, then the Schur multiplier is non-zero. We give a direct proof of an upper bound for the dimension of the Schur multiplier as a function of class and the minimum number of generators of the algebra. We then compare this bound with another known bound.