The classification of 4-dimensional Leibniz algebras

Cañete, E.M.; Khudoyberdiyev, A. Kh.

doi:10.48550/arxiv.1301.7665

Cited by 3 publications

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“…The second and strongest reason consists of the interactions and applications of Leibniz algebras in several areas of mathematics and mathematical physics such as: classical or non-commutative differential geometry, vertex operator algebras, the Godbillon-Vey invariants for foliations, integrable systems, representation theory, etc. For additional explanations and motivations we refer to [6,7,8,10,11,12,13,14,15,16,18,19,20,21,22,26,27,28] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…If L = {0}, then the GE-problem asks for the classification of all Leibniz algebra structures on an arbitrary vector space E. This is one of the problems that have been extensively studied during the last years and it is very difficult for vector spaces of large dimension: the classification of all Leibniz algebras is known up to dimension 4 and this result was obtained recently [12]. For this reason, from now on we will assume that L = {0}.…”

Section: Introductionmentioning

confidence: 99%

“…The explicit computation of the classifying object GHL 2 (L, g) is a challenging and very difficult problem. The first key step in decomposing this object is suggested by the compatibility condition(12) of Lemma 2.2: it shows that two different Leibniz algebra structures [−, −] g and [−, −] ′ g on g give two different equivalence classes (orbits) of the relation ≈ on CS(L, g). Let us fix [−, −] g a Leibniz algebra structure on g and denote by CS [−, −]g (L, g) the set of all local crossed systems of the Leibniz algebra L by the Leibniz algebra (g, [−, −] g ), i.e.…”

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confidence: 99%

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

Militaru

2014

Linear and Multilinear Algebra

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Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $\pi : E \to \mathfrak{L}$ an epimorphism of vector spaces with $ \mathfrak{g} = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on $E$ such that $\pi : E \to \mathfrak{L}$ is a morphism of Leibniz algebras: from a geometrical viewpoint this means 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 $E$ are classified by a global cohomological object ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ which is explicitly constructed. It is shown that ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ is the coproduct of all local cohomological objects $ {\mathbb H} {\mathbb L}^{2} \, \, (\mathfrak{L}, \, (\mathfrak{g}, [-,-]_{\mathfrak{g}}))$ that are classifying sets for all extensions of $\mathfrak{L}$ by all Leibniz algebra structures $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ on $\mathfrak{g}$. The second cohomology group ${\rm HL}^2 \, (\mathfrak{L}, \, \mathfrak{g})$ of Loday and Pirashvili appears as the most elementary piece among all components of ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$. Several examples are worked out in details for co-flag Leibniz algebras over $\mathfrak{L}$, i.e. Leibniz algebras $\mathfrak{h}$ that have a finite chain of epimorphisms of Leibniz algebras $\mathfrak{L}_n : = \mathfrak{h} \stackrel{\pi_{n}}{\longrightarrow} \mathfrak{L}_{n-1} \, \cdots \, \mathfrak{L}_1 \stackrel{\pi_{1}} {\longrightarrow} \mathfrak{L}_{0} := \mathfrak{L}$ such that ${\rm dim} ({\rm Ker} (\pi_{i})) = 1$, for all $i = 1, \cdots, n$.Comment: 20 pages: Continues arXiv:1307.254

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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

Militaru

2014

Linear and Multilinear Algebra

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“…Several classical theorems known in the context of Lie algebras were extended to Leibniz algebras, there exists a (co)homology theory for them, the classification of certain types of Leibniz algebras of a given (small) dimension was recently performed, their interaction with vertex operator algebras, the Godbillon-Vey invariants for foliations or differential geometry was highlighted. For more details and motivations we refer to [5], [6], [7], [9], [10], [11], [12], [13], [15], [16], [17], [18], [20], [21], [25], [26], [27] and the references therein. The starting point of this paper is the following question:…”

Section: Introductionmentioning

confidence: 99%

“…4-dimensional) Leibniz algebras was finished only recently in [13] (resp. [11]). For this reason, from now on we will assume that g = {0}.…”

Section: Introductionmentioning

confidence: 99%

Unified products for Leibniz algebras. Applications

Agore

Militaru

2013

Linear Algebra and its Applications

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Abstract. 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: HL 2 g (V, g) provides the classification up to an isomorphism that stabilizes g and HL 2 (V, g) will classify all such structures from the view point of the extension problem -here

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Second-Maximal Subalgebras of Leibniz Algebras

Bosko-Dunbar¹,

Dunbar²,

Stagg³

2016

Preprint

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In this work we study Leibniz algebras whose second-maximal subalgebras are ideals. We provide a classification based on solvability, nilpotency, and the size of the derived algebra. We give specific descriptions of those Leibniz algebras whose derived algebra has codimension zero or one. This includes several algebras which are only possible over certain fields.2010 Mathematics Subject Classification. 17D99.

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The classification of 4-dimensional Leibniz algebras

Cited by 3 publications

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

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

Unified products for Leibniz algebras. Applications

Second-Maximal Subalgebras of Leibniz Algebras

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