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.     

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.     

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.     

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.     

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.     

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.     

Schunck Classes of Soluble Leibniz Algebras

I set out the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algebras. Primitive Leibniz algebras come in pairs, one (Lie) symmetric, the other antisymmetric. A Schunck formation containing one member of a pair also contains the other. If ? is a Schunck formation and H is an ?-projector of the Leibniz algebra L, then H is intravariant in L. An example is given to show that the assumption that the Schunck class ? is a formation cannot be omitted.     

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.     

The Nilpotency Properties of the Leibniz Algebra Mn(C)D

We study the nilpotency properties of the Leibniz algebras constructed by means of D-mappings on the algebra of complex square matrices M _n (C). In particular, we obtain a criterion for nilpotency of these algebras in terms of the properties of a D-mapping. We prove also that the Leibniz algebras under consideration cannot be simple.     

Some Theorems on Leibniz n-Algebras from the Category Un(Lb)

We study the Leibniz n-algebra Un(∑),whose multiplication is defined viathe bracket of a Leibniz algebra ∑ as[x1,...,xn]=[x1,[...,[xn-2,[xn-1,xn]]...]].Weshow that Un(∑) is simple if and only if ∑ is a simple Lie algebra.An analog of Levi'stheorem for Leibniz algebras in Un(Lb) is established and it is proven that the Leibnizn-kernel of Un(Σ) for any semisimple Leibniz algebra Σ is the n-algebra Un(Σ).     

Ext groups in the category of bimodules over a simple Leibniz algebra

In this article, we generalize Loday and Pirashvili's [11] computation of the Ext-category of Leibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most of the arguments generalize easily, while the main new ingredient is the Feldvoss-Wagemann's cohomology vanishing theorem for semi-simple Leibniz algebras.     

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.     

Isomorphism classes and invariants of low-dimensional filiform Leibniz algebras

In the paper, we extend the result on classification of a subclass of filiform Leibniz algebras in low dimensions to dimensions seven and eight based on a technique used by Rakhimov and Bekbaev for classification of subclasses which arise from naturally graded non-Lie filiform Leibniz algebras. The class considered here arises from naturally graded filiform Lie algebras. It contains the class of filiform Lie algebras and consequently, by classifying this subclass, we again re-examine the classification result of filiform Lie algebras. The resulting list of filiform Lie algebras is compared with that given by Ancochéa-Bermúdez and Goze in 1988 and by Gómez, Jiménez-Merchán and Khakimdjanov in 1998.     

On Universal Central Extensions of Leibniz Algebras

We construct the endofunctor 𝔲𝔠𝔢 between the category of Leibniz algebras which assigns to a perfect Leibniz algebra its universal central extension, and we obtain the isomorphism 𝔲𝔠𝔢_Lie(𝔮_Lie) ? (𝔲𝔠𝔢_Leib(𝔮))_Lie, where 𝔮 is a perfect Leibniz algebra satisfying the condition [x, [x, y]] + [[x, y], x] = 0, for all x, y ∈ 𝔮. Moreover, we obtain several results concerning the lifting of automorphisms and derivations in a covering. We also study the relationship between the universal central extension of a semidirect product of perfect Leibniz algebras and the semidirect product of the universal central extension of both of them.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

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 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.