On classification of four-dimensional nilpotent Leibniz algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper, we give the classification of four-dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms and some other techniques to obtain our result.     

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.     

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.     

Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

In ([11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the Leibniz product gives rise to other types of invariance for a bilinear form defined on a Leibniz algebra: the left invariance, the right invariance. In this article, we study the structure of Leibniz algebras endowed with nondegenerate, symmetric, and left (resp. right) invariant bilinear forms. In particular, the existence of such a bilinear form on a Leibniz algebra 𝔏 gives rise to a new algebra structure ☆ on the underlying vector space 𝔏. In this article, we study this new algebra, and we give information on the structure of this type of algebras by using some extensions introduced in [11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]. In particular, we improve the results obtained in [22 Lin, J., Chen, Z. (2010). Leibniz algebras with pseudo-Riemannian bilinear forms. Front. Math. China 5(1):103–115.[Crossref], [Web of Science ®] , [Google Scholar]].     

Representations and T*-Extensions of Hom–Jordan–Lie Algebras

The purpose of this article is to study representations and T*-extensions of δ-hom–Jordan–Lie algebras. In particular, adjoint representations, trivial representations, deformations, and many properties of T*-extensions of δ-hom–Jordan–Lie algebras are studied in detail. Derivations and central extensions of δ-hom–Jordan–Lie algebras are also discussed as an application.     

T *-extension of n-Lie algebras

We introduce a new technique, called the T?*-extension, of constructing metric n-Lie algebras from arbitrary ones and obtain some important properties on this extension. Finally, we investigate the metric n-Lie algebras that are isometric to certain T?*-extensions.     

-extension of Lie triple systems

In this paper, we introduce the notion of T^*-extension of a Lie triple system. Then we show that T^*-extension is compatible with nilpotency, solvability, and it preserves in certain sense the decomposition properties. In addition, we investigate the equivalence of T^*-extensions using cohomology. Finally, we show that every finite-dimensional nilpotent metrised Lie triple system over an algebraically closed field is the T^*-extension of an appropriate quotient system.     

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).     

A Characterization of Nilpotent Leibniz Algebras

W. A. Moens proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper, using the definition of a Leibniz-derivation from Moens (2010), we show that a similar result for non-Lie Leibniz algebras is not true. Namely, we give an example of non-nilpotent Leibniz algebra that admits an invertible Leibniz-derivation. In order to extend the results of the paper by Moens (2010) for Leibniz algebras, we introduce a definition of a Leibniz-derivation of Leibniz algebras that agrees with Leibniz-derivation of the Lie algebra case. Further, we prove that a Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation of Definition 3.4. Moreover, the result that a solvable radical of a Lie algebra is invariant with respect to a Leibniz-derivation was extended to the case of Leibniz 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.     

Conservative algebras of 2-dimensional algebras,II

In 1990 Kantor defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n>1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of W(2). Also similar problems are solved for the algebra W₂ of all commutative algebras on the 2-dimensional vector space and for the algebra S₂ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.     

Isoclinism between pairs of n-Lie algebras and their properties

In this paper, we investigate the notion of isoclinism on a pair of n-Lie algebras, which forms an equivalence relation. In addition, we prove that each equivalence class contains a stem pair of n-Lie algebras, which has minimal dimension amongst the finite dimensional pairs of n-Lie algebras. Finally, some more results are obtained when two isoclinic pairs of n-Lie algebras are given.     

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.     

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.     

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.     

On the description of Leibniz algebras with nilindex <Emphasis Type="Italic">n</Emphasis>−3

We present the classification of a subclass of naturally graded Leibniz algebras. These n-dimensional Leibniz algebras have the characteristic sequence (n−3,3). For this purpose we use the software Mathematica.     

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E ₆, E ₇ and E ₈ which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of “good” mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.