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The description of complex solvable Leibniz algebras whose nilradical is a naturally graded filiform algebra is already known. Unfortunately, a mistake was made in that description. Namely, in the case where the dimension of the solvable…

We show that a solvable real rigid Lie algebra is not completelt rigid, by constructing an example of minimal dimension where the external torus is not spanned by $ad$-semisimple derivations over $\mathbb{R}$. We analyze the real forms of…

The purpose of this paper is to introduce and study nilpotent and filiform Hom-Lie algebras. Moreover, we extend Vergne and Khakimdjanov's approach to Hom-type algebras and provide a classification of filiform Hom-Lie algebras of dimension…

This paper is the first of a 3-part series that classifies the 5-dimensional Thurston geometries. The present paper (part 1 of 3) summarizes the general classification, giving the full list, an outline of the method, and some illustrative…

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…

We construct indefinite Einstein solvmanifolds that are standard, but not of pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form $\mathfrak{g}\rtimes_D\mathbb{R}$, where $\mathfrak{g}$ is a nilpotent Lie algebra and $D$ is…

We determine fundamental systems of invariants for complex solvable rigid Lie algebras having nonsplit nilradicals of characteristic sequence $(3,1,..,1)$, these algebras being the natural followers of solvable algebras having Heisenberg…

A Riemannian manifold (M,g) is said to be Einstein if its Ricci tensor satisfies ric(g) = cg, for some real number c. In the homogeneous case, a problem that is still open is the so called Alekseevskii Conjecture. This conjecture says that…

In this paper we investigate pre-derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three non-intersected families. We describe the pre-derivation of filiform…

We give a geometric classification of 4-dimensional superalgebras over an algebraic closed field.

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_{n,3} which contains the previously studied filiform nilpotent algebra n_{n-2,1} as a subalgebra but not as an ideal. Rather surprisingly it turns out that…

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension $15$ and decomposes into $3$ irreducible components…

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative…

In this study, we classify some soliton nilpotent Lie algebras and possible candidates in dimension 8 and 9 up to isomorphy. We focus on 1 < 2 < ::: < n type of derivations where n is the dimension of the Lie algebras. We present algorithms…

In math.DG/0312243 we developed a general classification scheme for metric Lie algebras, i.e. for finite-dimensional Lie algebras equipped with a non-degenerate invariant inner product. Here we determine all nilpotent Lie algebras l with…

We classify (possibly non commutative) algebras of low rank over a domain R. We first review results for algebras of rank 2 and for finite-dimensional division algebras over the real numbers. These results motivate us to consider which…

In the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations,…

We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures,…

In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is…

We present a list of all isomorphism classes of nonsolvable Lie algebras of dimension less than 7 over a finite field.