Related papers: Born Lie algebras

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures…

In this paper we prove that in classifying of complex filiform Leibniz algebras, for which its naturally graded algebra is non-Lie algebra, it suffices to consider some special basis transformations. Moreover, we establish a criterion…

A new kind of graded Lie algebra (we call it $Z_{2,2}$ graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable bose subspace of the $Z_{2,2}$ graded Lie algebra and using relevant…

We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic…

The main goal is to classify 4-dimensional real Lie algebras $\g$ which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore…

The classification of complex of real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example the nilpotent Lie algebras are classified only up to the dimension 7. Moreover, to recognize a given…

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,…

We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic…

In this paper, we classify solvable Lie algebras of dimensions $\leq 8$ endowed with a nondegenerate invariant symmetric bilinear form over an algebraically closed field. This classification (up to isometrically isomorphisms) is mainly…

All bialgebra structures on twodimensional Galilei algebra are classified. The corresponding Lie-Poisson structures on Galilei group are found.

It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to…

We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms faithfully flat cohomology over an arbitrary ring…

We propose a simple definition of a Born geometry in the framework of K\"unneth geometry. While superficially different, this new definition is equivalent to the known definitions in terms of para-quaternionic or generalized geometries. We…

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 determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie…

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, S\'erie I, (2001) 763-768) with the canonical linear…

Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras…

Some forms of Lie algebras of types E_6, E_7, and E_8 are constructed using the exterior cube of a rank 9 finitely generated projective module.

The structure of a solvable Lie groups 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…

We classify all integrable complex structures on 6-dimensional Lie algebras of the form $\mathfrak{g}\times\mathfrak{g}$.