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     

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.     

Classification of solvable Leibniz algebras with null-filiform nilradical

In this article we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra as well.     

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.     

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.      

Right and left solvable extensions of an associative Leibniz algebra

A sequence of nilpotent Leibniz algebras denoted by N_n,18 is introduced. Here n denotes the dimension of the algebra defined for n≥4; the first term in the sequence is ?₁₈ in the list of four-dimensional nilpotent Leibniz algebras introduced by Albeverio et al. [4 Albeverio, S., Omirov, B. A., Rakhimov, I. S. (2006). Classification of 4-dimensional nilpotent complex Leibniz algebras. Extr. Math. 21(3):197–210. [Google Scholar]]. Then all possible right and left solvable indecomposable extensions over the field ? are constructed so that N_n,18 serves as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

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.     

The Classification of Naturally Graded p-Filiform Leibniz Algebras

In the present article the classification of n-dimensional naturally graded p-filiform (1 ≤ p ≤ n ? 4) Leibniz algebras is obtained. A splitting of the set of naturally graded Leibniz algebras into the families of Lie and non Lie Leibniz algebras by means of characteristic sequences (isomorphism invariants) is proved.     

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Solvable extensions of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3

For sequences of naturally graded quasi-filiform Leibniz algebras of second type ?¹ and ?³ introduced by Camacho et al., all possible right and left solvable indecomposable extensions over the field ? are constructed so that these algebras serve as the nilradicals of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Classification of three-dimensional solvable <Emphasis Type="Italic">p</Emphasis>-adic Leibniz algebras

The present paper is devoted to the study of low dimensional Leibniz algebras over the field of p-adic numbers. The classification up to isomorphism of three-dimensional Lie algebras over the integer p-adic numbers is already known [8]. Here, we extend this classification to solvable Lie and non-Lie Leibniz algebras over the field of p-adic numbers.     

On Solvability and Nilpotency of Leibniz n-Algebras

The concepts of solvable and nilpotent Leibniz n-algebra are introduced, and classical results of solvable and nilpotent Lie algebras theory are extended to Leibniz n-algebras category. A homological criterion similar to Stallings Theorem for Lie algebras is obtained in Leibniz n-algebras category by means of the homology with trivial coefficients of Leibniz n-algebras.     

Some notes on the second homology of Leibniz algebras

Some properties of the second homology and cover of Leibniz algebras are established. By constructing a stem cover, the second Leibniz homology and cover of abelian, Heisenberg Lie algebras and cyclic Leibniz algebras are described. Also, for the dimension of a non-cyclic nilpotent Leibniz algebra L, we obtain dim(HL₂(L))≥2.     

Graded extended Lie-type algebras

The class of extended Lie-type algebras contains the ones of associative algebras, Lie algebras, Leibniz algebras, dual Leibniz algebras, pre-Lie algebras, and Lie-type algebras, etc. We focus on the class of extended Lie-type algebras graded by an Abelian group G and study its structure, by stating, under certain conditions, a second Wedderburn-type theorem for this class of algebras.     

Conservative algebras of 2-dimensional algebras,II

In 1990 Kantor defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n>1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of W(2). Also similar problems are solved for the algebra W₂ of all commutative algebras on the 2-dimensional vector space and for the algebra S₂ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.     

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).      

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Solvable Lie Algebras with Nilradical (Q)2n+1 and Their Casimir Invariants

The finite-dimensional indecomposable solvable Lie algebras (s) with (Q)2n+1 as their nilradical are studied and classified and their Casimir invariants are calculated.It turns out that the dimension of (s) is at most dim(Q)2n+1+2.