Structure Of Leibniz Algebras Research Articles

On the derivations of cyclic Leibniz algebras

Let $L$ be an algebra over a field $F$. Then $L$ is called a left Leibniz algebra, if its multiplication operation $[-,-]$ additionally satisfies the so-called left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of a Leibniz algebra $L$ is called a derivation of an algebra $L$, if $f([a,b])=[f(a),b]+[a,f(b)]$ for all elements $a,b\in L$. It is well known that the set of all derivations $\mathrm{Der}(L)$ of a Leibniz algebra $L$ is a subalgebra of the Lie algebra $\mathrm{End}_{F}(L)$ of all linear transformations of an algebra $L$. The algebras of derivations of Leibniz algebras play an important role in the study of structure of Leibniz algebras. Their role is similar to that played by groups of automorphisms in the study of group structure.
 In this paper, a complete description of the algebra of derivations of nilpotent cyclic Leibniz algebra is obtained. In particular, it was proved that this algebra is metabelian and supersoluble Lie algebra, and its dimension is equal to the dimension of an algebra $L$.

Methods of group theory in Leibniz algebras: some compelling results

The theory of Leibniz algebras has been developing quite intensively. Most of the results on the structural features of Leibniz algebras were obtained for finite-dimensional algebras and many of them over fields of characteristic zero. A number of these results are analogues of the corresponding theorems from the theory of Lie algebras. The specifics of Leibniz algebras, the features that distinguish them from Lie algebras, can be seen from the description of Leibniz algebras of small dimensions. However, this description concerns algebras over fields of characteristic zero. Some reminiscences of the theory of groups are immediately striking, precisely with its period when the theory of finite groups was already quite developed, and the theory of infinite groups only arose, i.e., with the time when the formation of the general theory of groups took place. Therefore, the idea of using this experience naturally arises. It is clear that we cannot talk about some kind of similarity of results; we can talk about approaches and problems, about application of group theory philosophy. Moreover, every theory has several natural problems that arise in the process of its development, and these problems quite often have analogues in other disciplines. In the current survey, we want to focus on such issues: our goal is to observe which parts of the picture involving a general structure of Leibniz algebras have already been drawn, and which parts of this picture should be developed further.

On the influence of ideals and self-idealizing subalgebras on the structure of Leibniz algebras

The subalgebra A of a Leibniz algebra L is self-idealizing in L, if A = IL (A) . In this paper we study the structure of Leibniz algebras, whose subalgebras are either ideals or self-idealizing. More precisely, we obtain a description of such Leibniz algebras for the cases where the locally nilpotent radical is Abelian non-cyclic, non-Abelian noncyclic, and cyclic of dimension 2.

Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.

On Theta Pairs and Theta Completions for Proper Subalgebras in Leibniz Algebras

In this paper, we investigate the structure of Leibniz algebras, using the concepts of the $$\theta $$ -pair and the $$\theta $$ -completion that are analogous to the corresponding ones in group theory and Lie algebras. In fact, using these concepts for proper subalgebras, we give some characterizations for the solvability and the supersolvability of Leibniz algebras.

On the structure of Leibniz algebras, whose subalgebras are ideals or core-free

On the structure of Leibniz algebras, whose subalgebras are ideals or core-free

On the structure of Leibniz algebras whose subalgebras are ideals or core-free

An algebra \(L\) over a field \(F\) is said to be a Leibniz algebra (more precisely, a left Leibniz algebra) if it satisfies the Leibniz identity: \([[a, b], c] = [a, [b, c]] - [b, [a, c]]\) for all \(a, b, c \in L\). Leibniz algebras are generalizations of Lie algebras. A subalgebra \(S\) of a Leibniz algebra \(L\) is called a core-free, if \(S\) does not include a non-zero ideal. We study the Leibniz algebras, whose subalgebras are either ideals or core-free.

Leibniz Algebras Associated with Representations of Euclidean Lie Algebra

In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $\mathfrak {e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by I) as a right $\mathfrak {e}(2)$-module is associated to representations of $\mathfrak {e}(2)$ in $\mathfrak {sl}_{2}({\mathbb {C}})\oplus \mathfrak {sl}_{2}({\mathbb {C}}), \mathfrak {sl}_{3}({\mathbb {C}})$ and $\mathfrak {sp}_{4}(\mathbb {C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${\mathfrak {e(n)}}$ as its liezation I being an (n + 1)-dimensional right ${\mathfrak {e(n)}}$-module defined by transformations of matrix realization of $\mathfrak {e(n)}$. Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $\mathfrak {D}_{k}$ and describe the structure of Leibniz algebras with corresponding Lie algebra $\mathfrak {D}_{k}$ and with the ideal I considered as a Fock $\mathfrak {D}_{k}$-module.

Leibniz Algebras Associated to Extensions of sl 2

In this work, we investigate the structure of Leibniz algebras whose associated Lie algebra is a direct sum of sl 2 and the solvable radical. In particular, we obtain the description of such algebras when: the ideal generated by the squares of elements of a Leibniz algebra is irreducible over sl 2 and when the dimension of the radical is equal to two.

The global extension problem, co-flag and metabelian Leibniz algebras

Let be a Leibniz algebra, be a vector space and be an epimorphism of vector spaces with . The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on such that is a morphism of Leibniz algebras: from a geometrical viewpoint this is to give the decomposition of the groupoid of all such structures in its connected components and to indicate a point in each component. All such Leibniz algebra structures on are classified by a global cohomological object which is explicitly constructed. It is shown that is the coproduct of all local cohomological objects which are classifying sets for all extensions of by all Leibniz algebra structures on . The second cohomology group of Loday and Pirashvili appears as the most elementary piece among all components of . Several examples are worked out in details for co-flag Leibniz algebras over , i.e. Leibniz algebras that have a finite chain of epimorphisms of Leibniz algebras such that , for all . Metabelian Leibniz algebras are introduced, described and classified using pure linear algebra tools.

Unified products for Leibniz algebras. Applications

Let g be a Leibniz algebra and E a vector space containing g as a subspace. All Leibniz algebra structures on E containing g as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: HLg2(V,g) provides the classification up to an isomorphism that stabilizes g and HL2(V,g) will classify all such structures from the view point of the extension problem – here V is a complement of g in E. A general product, called the unified product, is introduced as a tool for our approach. The crossed (resp. bicrossed) products between two Leibniz algebras are introduced as special cases of the unified product: the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem. The description and the classification of all complements of a given extension g⊆E of Leibniz algebras are given as a converse of the factorization problem. They are classified by another cohomological object denoted by HA2(h,g|(▹,◃,↼,⇀)), where (▹,◃,↼,⇀) is the canonical matched pair associated to a given complement h. Several examples are worked out in details.

An algorithm for the classification of 3-dimensional complex Leibniz algebras

We propose an algorithm using Gröbner bases that decides in terms of the existence of a non singular matrix P if two Leibniz algebra structures over a finite dimensional C-vector space are representative of the same isomorphism class.We apply this algorithm in order to obtain a reviewed classification of the 3-dimensional Leibniz algebras given by Ayupov and Omirov. The algorithm has been implemented in a Mathematica notebook.

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

Motivated by Kapranov's discovery of an sh Lie algebra structure on the tangent complex of a K\"{a}hler manifold and Chen-Sti\'{e}non-Xu's construction of sh Leibniz algebras associated with a Lie pair, we find a general method to construct sh Leibniz algebras. Let $\mathcal{A}$ be a commutative dg algebra. Given a derivation of $\mathcal{A}$ valued in a dg module $\Omega$, we show that there exist sh Leibniz algebra structures on the dual module of $\Omega$. Moreover, we prove that this process establishes a functor from the category of dg module valued derivations to the category of sh Leibniz algebras over $\mathcal{A}$.