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Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between…
One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous…
Leibniz algebras are certain generalization of Lie algebras. It is natural to generalize concepts in Lie algebras to Leibniz algebras and investigate whether the corresponding results still hold. In this paper we introduce the notion of…
Dialgebras are generalizations of associative algebras which give rise to Leibniz algebras instead of Lie algebras. In this paper we study super dialgebras and Leibniz superalgebras, which are $\z_2$-graded dialgebras and Leibniz algebras.…
We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric,…
The aim of this article is to obtain variations on the classical theorems of Schur and Baer on finiteness of commutator subgroups, valid in the contexts of Lie algebras and Leibniz algebras over a field. Using non-abelian tensor products…
This paper concerns the study of Leibniz algebras, a natural generalization of Lie algebras, from the perspective of centralizers of elements. We study conditions on Leibniz algebras under which centralizers of all elements are ideals. We…
This paper gives an overview of some basic properties of Leibniz algebras. Some of the results were known earlier, but in the article they are accompanied by new simple proofs. Some of the results are new. The article can be viewed as a…
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…
This paper introduces and investigates the structure of $\delta$-Leibniz algebras, which serve as a parametric generalization of classical Leibniz algebras defined by a scalar $\delta$. The authors define $\delta$-Lie algebras, $\delta$-Lie…
Since Leibniz algebras were introduced by Loday as a generalization of Lie algebras, there has been a lot of interest in which results of the latter extend to the former. Cyclic algebras, those generated by one element, are a useful tool…
I give a simpler proof of the generalisation of Engel's Theorem to Leibniz algebras.
Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of this paper is to generalise these results in two different…
In this article we present a generalization of a Leibniz's geometrical theorem and an application of it.
Left and right "generalized Schur algebras", previously introduced by the author, are defined and analyzed. Filtrations of these algebras lead, in most cases, to parameterizations of the their irreducible representations over fields of…
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 and several results seem to be new.
In this note, we observe a relation between dialgebras (in particular, Leibniz algebras) and conformal algebras. The purpose is to show how the methods of conformal algebras help solving problems on dialgebras, and, conversely, how the…
Leibniz algebras are non-antisymmetric generalizations of Lie algebras that have attracted substantial interest due to their close relation with the latter class. A Leibniz algebra $A$ is called perfect if it coincides with its derived…
Leibniz algebras generated by one element, called cyclic, provide simple and illuminating examples of many basic concepts. It is the purpose of this paper to illustrate this fact.
In this paper we study the notion of isoclinism on Lie-central extensions of Leibniz algebras, this yields to introduce the concept of Lie-isoclinic Leibniz algebras. We provide several equivalent conditions under which Leibniz algebras are…