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).      

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 extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left‐invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one‐dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.     

Classification of Nilsoliton metrics in dimension seven

The aim of this paper is to classify Ricci soliton metrics on 7-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in Fernández Culma (2012). To this end, we use the classification of 7-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski (2010, 2012) and of Yuri Nikolayevsky (2011). Of the 9 one-parameter families and 140 isolated 7-dimensional indecomposable real nilpotent Lie algebras, we have 99 nilsoliton metrics given in an explicit form and 7 one-parameter families admitting nilsoliton metrics.Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.     

Rings Whose Nilpotent Elements form a Levitzki Radical Ring

In this article, we first discuss the relations among JH^r = 0, JH^r?1·H = 0, and JH^r·x = 0. Then we give a counterexample to the question mentioned in the Remarks of [3 Cheng, C. C., Sakkalis, T., Wang, S. S. S. (1994). A case of the Jacobian conjecture. J. Pure and Applied Algebra 96:15–18.[Crossref], [Web of Science ®] , [Google Scholar]] and prove the equivalence among JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r)) = 0, JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r?1))·H(x^(r)) = 0, and JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r))·x^(r) = 0. Finally, we give partial answer to Conjecture 2 in [4 Connell, E., Zweibel, J. (1991). Exact and coexact matrices. J. Algebra 142:110–117.[Crossref], [Web of Science ®] , [Google Scholar]].     

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 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     

On Sums of Coefficients of Products of Polynomials

We study the nilpotency of the sums of all coefficients of some sorts of products of polynomials over reversible, IFP, and NI rings, and introduce an SCN ring as a generalization. We characterize SCN rings in relation with related ring properties, and also provide several useful properties and ring extensions of SCN rings.