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We extend the classical construction of solvable Lie algebras from a nilradical to compatible Lie algebras. Since the sum of nilpotent ideals may fail to be nilpotent, we replace the usual nilradical by a \emph{special nilradical} that…

This paper shows that a Poisson algebra is nilpotent if and only if it is both associative and Lie nilpotent and examines various properties of the nilradical and the solvable radical. It introduces a basic Frattini theory for dialgebras…

We describe solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum length.

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive…

Every real simple non-compact Lie algebra not isomorphic to $\mathfrak{so}(1,n)$ contains a unique standard parabolic subalgebra whose nilradical is a generalized Heisenberg algebra. Here we discuss the associated parabolic geometries and…

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. In particular, we describe the complex and the real case of the indecomposable…

We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian…

In this paper we introduce the notion of pure non-characteristically nilpotent Lie algebra and under a condition we prove that a complex maximal extension of a finite-dimensional pure non-characteristically nilpotent Lie algebra is…

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the…

In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci…

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence…

This paper is a contribution to the development of the non associative algebras theory. More precisely, this work deals with the classification of the complex 4-dimensional Leibniz algebras. Note that the classification of 4-dimensional…

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new…

The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded quasi-filiform algebra and the complemented space to the…

We propose the study and description of the structure of complex Lie algebras with nilradical a nilpotent Lie algebra of type 2 by using sl2(C)-representation theory. Our results will be applied to review the classification given in [1] (J.…

We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph. We then use properties of the Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie…

This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$.…

We classify all nonnilpotent, solvable Leibniz algebras with the property that all proper subalgebras are nilpotent. This generalizes the work of Stitzinger and Towers in Lie algebras. We show several examples which illustrate the…

The concept of breadth has been used in the classification of p-groups and nilpotent Lie algebras. In this paper, we investigate this notion for finite-dimensional solvable Lie algebras. Our main focus is to characterize solvable Lie…

In this paper, we describe the automorphisms of solvable Leibniz algebras with null-filiform nilradical. Moreover we describe the automorphisms of solvable Leibniz algebras with naturally graded non-Lie filiform nilradicals, whose the…