Solvable Leibniz algebras with naturally graded non-Lie p-filiform nilradicals

In this paper solvable Leibniz algebras with naturally graded non-Lie p-filiform (n?p≥4) nilradical and with one-dimensional complemented space of nilradical are described. Moreover, solvable Leibniz algebras with abelian nilradical and extremal (minimal, maximal) dimensions of complemented space nilradical are studied. The rigidity of solvable Leibniz algebras with abelian nilradical and maximal dimension of its complemented space is proved.     

Solvable Leibniz Algebras with Heisenberg Nilradical

All solvable Lie algebras with Heisenberg nilradical have already been classified. We extend this result to a classification of solvable Leibniz algebras with Heisenberg nilradical. As an example, we show the complete classification of all real or complex Leibniz algebras whose nilradical is the 3-dimensional Heisenberg algebra.     

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type ℒ2

For a sequence of the naturally graded quasi-filiform Leibniz algebra of second type ?² introduced by Camacho, Gómez, González and Omirov, all possible right and left solvable indecomposable extensions over the field ? are constructed so that the algebra serves as the nilradical of the corresponding solvable Leibniz algebras we find in the paper.     

Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicals

Abstract

In this article, solvable Leibniz algebras, whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to the nilradical is proved.

Communicated by K. C. Misra     

Einstein solvmanifolds with a simple Einstein derivation

The structure of a solvable Lie group admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).      

Description of some classes of Leibniz algebras

In this paper we describe the isomorphism classes of finite-dimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl₂ and a three-dimensional solvable ideal. We choose a basis of the isomorphism classes’ representatives and give explicit multiplication tables.     

Gröbner-Shirshov bases for free Gelfand-Dorfman-Novokov algebras and for right ideals of free right Leibniz algebras

In this paper, we first found a magmatic (i.e., absolutely non-associative) Gröbner-Shirshov basis of a free Gelfand-Dorfman-Novikov algebra GDN(X) such that the corresponding set of irreducible magmatic words is the Dzhumadildaev-Löfwall linear basis of the GDN(X). Then, we prove a Composition-Diamond lemma for right ideals of a free right Leibniz algebra Lei(X).     

A Comment on the Integration of Leibniz Algebras

In this note we point out that the definition of the universal enveloping dialgebra for a Leibniz algebra is consistent with the interpretation of a Leibniz algebra as a generalization not of a Lie algebra, but of the adjoint representation of a Lie algebra. From this point of view, the formal integration problem of Leibniz algebras is, essentially, trivial.     

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

Einstein solvmanifolds with free nilradical

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the Abelian and the two-step ones, there are only six others: is a free p-step Lie algebra on m generators). The reason for that is the inequality-type restrictions on the eigenvalue type of an Einstein nilradical obtained in the paper.      

A Characterization of Nilpotent Leibniz Algebras

W. A. Moens proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper, using the definition of a Leibniz-derivation from Moens (2010), we show that a similar result for non-Lie Leibniz algebras is not true. Namely, we give an example of non-nilpotent Leibniz algebra that admits an invertible Leibniz-derivation. In order to extend the results of the paper by Moens (2010) for Leibniz algebras, we introduce a definition of a Leibniz-derivation of Leibniz algebras that agrees with Leibniz-derivation of the Lie algebra case. Further, we prove that a Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation of Definition 3.4. Moreover, the result that a solvable radical of a Lie algebra is invariant with respect to a Leibniz-derivation was extended to the case of Leibniz algebras.     

ON LIE ALGEBRAS WHOSE NILRADICAL IS (n — p)-FILIFORM

We prove first that every (n — p)-filiform Lie algebra, p ≤ 3, is the nilradical of a solvable, nonnilpotent rigid Lie algebra. We also analize how this result extends to (n — 4)-filiform Lie algebras. For this purpose, we give a classificaction of these algebras and then determine which of the obtained classes appear as the nilradical of a rigid algebra.     

Einstein solvmanifolds attached to two-step nilradicals

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra which can serve as the nilradical of an Einstein metric solvable Lie algebra is called an Einstein nilradical. We give a classification of two-step nilpotent Einstein nilradicals with two-dimensional center. Informally, the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity. We also show that the dual to a two-step Einstein nilradical is not in general an Einstein nilradical.     

On Behavior of Solvable Ideals of Lie Algebras Under Outer Derivations

Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log₂ p.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

<Emphasis Type="Italic">n</Emphasis>-Tuple algebras of associative type

We study properties of n-tuple algebras of associative type. We show that the nilpotency of an n-tuple algebra of associative type is determined by the nilpotency of each element. In addition, we characterize the nilpotency of an n-tuple algebra of associative type in terms of the trace function. In the final part of the paper, we show that a homogeneously semisimple n-tuple algebra of associative type is the direct sum of two-sided ideals each of which is a homogeneously simple n-tuple algebra of associative type.     

A construction of quasi-hereditary endomorphism algebras简

In this article, we consider endomorphism algebras of direct sums of some local left ideals over a local algebra and give a construction of quasi-hereditary algebras.     

On the nilpotency and decomposition of Lie-type algebras

In the paper, an analog of the Engel theorem for graded algebras admitting a Lie-type module is proved. Moreover, it is shown that every semisimple algebra of associative type with ordered grading and one-dimensional grading subspaces is the direct sum of two-sided ideals that are simple algebras.