A Leibniz Algebra Structure on the Second Tensor Power

Symmetry Integrability and Geometry-methods and Applications, 2013

This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear pairing taking values in the Leibniz kernel.

Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff subcategory of Lie algebras, called Lie-central extensions. We construct a Lie-homology of Leibniz algebras and obtain a six-term exact homology sequence associated to a Lie-central extension. This sequence, together with the relative commutators, allows us to characterize several classes of Lie-central extensions, such as Lie-trivial, Lie-stem and Lie-stem cover, to introduce and characterize Lie-unicentral, Lie-capable, Lie-solvable and Lie-nilpotent Leibniz algebras.

2018

From the theory of finite dimensional Lie algebras it is known that every finite dimensional Lie algebra is decomposed into a semidirect sum of semisimple subalgebra and solvable radical. Moreover, due to work of Mal’cev the study of solvable Lie algebras is reduced to the study of nilpotent ones. For the finite dimensional Leibniz algebras the analogues of the mentioned results are not proved yet. In order to get some idea how to establish the results we examine the Leibniz algebra for which the quotient algebra with respect to the ideal generated by squares elements of the algebra (denoted by I) is a semidirect sum of semisimple Lie algebra and the maximal solvable ideal. In this paper the class of complex Leibniz algebras, for which quotient algebras by the ideal I are isomorphic to the semidirect sum of the algebra sl2 and two-dimensional solvable ideal R, are described. Mathematics Subject Classification 2010: 17A32, 17B30.

Communications in Algebra, 2019

In the theories of groups and Lie algebras, investigations of the properties of the non-abelian tensor product and their relations to the second homology groups are worthwhile. It is the purpose of the present paper to exhibit such investigations about the non-abelian tensor product of Leibniz algebras. The isomorphism between the non-abelian tensor square and non-abelian exterior square of a Lie algebra L, will enable us to set a simple connection between HL 2 ðLÞ and H 2 ðLÞ. Furthermore, we shall relate the concepts of capability and solvability of a Leibniz algebra to its tensor square. Finally, we give an upper bound for the dimension of the nonabelian tensor square and the second homology of a nilpotent Leibniz algebra in terms of the dimension of its center and derived subalgebra.

arXiv: K-Theory and Homology, 2019

A Ronco algebra is a Leibniz algebra satisfying the identity: $$[[x,x],y]=0.$$ Based on properties of Leibniz homology, we give a proof an old and unpublished result of Maria Ronco, which describes free objects in these class of algebras. We also establish a relationship between Ronco algebras and so called Lie $\mu$-algebras extending our previous result.

Journal of Homotopy and Related Structures

Lie algebras and groups equipped with a multiplication $$\mu $$μ satisfying some compatibility properties are studied. These structures are called symmetric Lie $$\mu $$μ-algebras and symmetric $$\mu $$μ-groups respectively. An equivalence of categories between symmetric Lie $$\mu $$μ-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie $$\mu $$μ-groups and finite dimensional symmetric Leibniz algebras.

Contemporary Mathematics, 2019

In this paper we define the basic concepts for left or right Leibniz algebras and prove some of the main results. Our proofs are often variations of the known proofs but several results seem to be new.

2006

Abstract. The “coquecigrue ” problem for Leibniz algebras is that of finding an appropriate generalization of Lie’s third theorem, that is, of finding a generalization of the notion of group such that Leibniz algebras are the corresponding tangent algebra structures. The difficulty is determining exactly what properties this generalization should have. Here we show that Lie racks, smooth left distributive structures, have Leibniz algebra structures on their tangent spaces at certain distinguished points. One way of producing racks is by conjugation in digroups, a generalization of group which is essentially due to Loday. Using semigroup theory, we show that every digroup is a product of a group and a trivial digroup. We partially solve the coquecigrue problem by showing that to each Leibniz algebra that splits over an ideal containing its ideal generated by squares, there exists a special type of Lie digroup with tangent algebra isomorphic to the given Leibniz algebra. The general c...

Publicationes Mathematicae Debrecen, 2020

In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Liederivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ID *-Lie-derivations. A ID *-Lie-derivation of a Leibniz algebra g is a Lie-derivation of g in which the image is contained in the second term of the lower Lie-central series of g, and that vanishes on Lie-central elements. We provide an upperbound for the dimension of the Lie algebra ID Lie * (g) of ID *-Lie-derivation of g, and prove that the sets ID Lie * (g) and ID Lie * (q) are isomorphic for any two Lie-isoclinic Leibniz algebras g and q.

Bulletin of the Belgian Mathematical Society - Simon Stevin

A non-abelian tensor product for Leibniz n-algebras is introduced as a generalization of the non-abelian tensor product for Leibniz algebras introduced by Kurdiani and Pirashvili. We use it to construct the universal central extension of a perfect Leibniz n-algebra.