Leibniz algebras with pseudo-Riemannian bilinear forms

M. Bordemann has studied non-associative algebras with nondegenerate associative bilinear forms. In this paper, we focus on pseudo-Riemannian bilinear forms and study pseudo-Riemannian Leibniz algebras, i.e., Leibniz algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. We give the notion and some properties of T*-extensions of Leibniz algebras. In addition, we introduce the definition of equivalence and isometrical equivalence for two T*-extensions of a Leibniz algebra, and give a sufficient and necessary condition for the equivalence and isometrical equivalence.     

Classification of Some Solvable Leibniz Algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension ≤ 8 with one dimensional derived subalgebra. We use the canonical forms for the congruence classes of matrices of bilinear forms to obtain our result. Our approach can easily be extended to classify these algebras of higher dimensions. We also revisit the classification of three dimensional non-Lie solvable (left) Leibniz algebras.     

The Classification of Non-Characteristically Nilpotent Filiform Leibniz Algebras

In this paper we investigate the derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension is decomposed into three non-intersected families. We found sufficient conditions under which filiform Leibniz algebras of the first family are characteristically nilpotent. Moreover, for the first family we classify non-characteristically nilpotent algebras by means of Catalan numbers. In addition, for the rest two families of filiform Leibniz algebras we describe non-characteristically nilpotent algebras, i.e., those filiform Leibniz algebras which lie in the complementary set to those characteristically nilpotent.     

完备Leibniz代数的性质及其低维分类

本文研究了完备Leibniz代数的性质及低维分类.利用Leibniz代数中平方元生成的双边理想,获得了小于五维的完备Leibniz代数完整的分类,以及五维时一类特殊情况下完备Leibniz代数的分类,从而推广了Leibniz代数的结构理论.     

Criteria for solvability and supersolvability in Leibniz algebras

Leibniz algebras are certain generalization of Lie algebras. Recently, analyzing the structure of subalgebras, David Towers gave some criteria for the solvability and supersolvability of Lie algebras. In this paper we define analogues concepts for Leibniz algebras and extend some of these results on solvability and supersolvability to that of Leibniz algebras.     

形变Schrédinger-Virasoro 代数的非退化对称不变双线性型

本文确定了形变Schrödinger-Virasoro 代数的非退化对称不变双线性型, 并借助此类Lie 代数上的二上同调群, 确定了相应的Leibniz 二上同调群.     

Suite spectrale d'une extension d'algèbres de Leibniz

Leibniz homology is a non-commutative homology theory for Lie algebras which can be extended to a larger class of algebras: the Leibniz algebras. We construct a “Leibniz version” of the Hochschild-Serre spectral sequence.     

Classifying Several Classes of Leibniz Algebras

We extend results related to maximal subalgebras and ideals from Lie to Leibniz algebras. In particular, we classify minimal non-elementary Leibniz algebras and Leibniz algebras with a unique maximal ideal. In both cases, there are types of these algebras with no Lie algebra analogue. We also give a classification of E-Leibniz algebras which is very similiar to its Lie algebra counterpart. Note that a classification of elementary Leibniz algebras has been shown in Batten Ray et al. (2011).     

Conformal representations of Leibniz algebras

We study the embedding construction of Lie dialgebras (Leibniz algebras) into conformal algebras. This construction leads to the concept of a conformal representation of Leibniz algebras. We prove that each (finite-dimensional) Leibniz algebra possesses a faithful linear representation (of finite type). As a corollary we give a new proof of the Poincaré-Birkhoff-Witt theorem for Leibniz algebras.     

三维Leibniz代数的分类

Leibniz代数是比Lie代数更广泛的一类代数,它通常不满足反交换性.在这篇文章里我们确定了维数等于3的Leibniz代数的同构类.     

微分算子Lie代数上的Leibniz 2-上循环

中心扩张问题在Leibniz代数的研究中起着非常重要的作用,因此有许多文章研究各种各样Leibniz代数的中心扩张问题.在这篇文章里,我们确定了微分算子Lie代数上的所有一维Leibniz中心扩张.     

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.     

Leibniz Homology of Extended Lie Algebras

Leibniz homology is a noncommutative homology theory for Lie algebras. In this paper, we compute low-dimensional Leibniz homology of extended Lie algebras.     

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.     

On non-Abelian Leibniz cohomology

In this note, by using a generalized notion of the Leibniz algebra of derivations, we present the constructions of the zero, first, and second non-Abelian Leibniz cohomologies with coefficients in crossed modules, which generalize the classical zero, first, and second Leibniz cohomology. For Lie algebras we compare the non-Abelian Leibniz and Lie cohomologies. We describe the second non-Abelian Leibniz cohomology via extensions of Leibniz algebras by crossed modules.     

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.     

Classification of a subclass of low-dimensional complex filiform Leibniz algebras

We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-Lie filiform Leibniz algebras. The isomorphism criteria in terms of invariant functions are given.     

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.     

A New Approach to Leibniz Bialgebras

A study of Leibniz bialgebras arising naturally through the double of Leibniz algebras analogue to the classical Drinfeld’s double is presented. A key ingredient of our work is the fact that the underline vector space of a Leibniz algebra becomes a Lie algebra and also a commutative associative algebra, when provided with appropriate new products. A special class of them, the coboundary Leibniz bialgebras, gives us the natural framework for studying the Yang-Baxter equation (YBE) in our context, inspired in the classical Yang-Baxter equation as well as in the associative Yang-Baxter equation. Results of the existence of coboundary Leibniz bialgebra on a symmetric Leibniz algebra under certain conditions are obtained. Some interesting examples of coboundary Leibniz bialgebras are also included. The final part of the paper is dedicated to coboundary Leibniz bialgebra structures on quadratic Leibniz algebras.