A cohomological characterization of Leibniz central extensions of Lie algebras

Hu, Naihong; Pei, Yufeng; Dong, Lixin

doi:10.1090/s0002-9939-07-08985-x

Cited by 30 publications

(30 citation statements)

References 22 publications

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“…This is what we do in Section 2. For further reference on the extension problem for Lie algebras, in the abelian or non-abelian case, we refer to [3], [4], [9], [11], [13], [24].…”

Section: Extending Structures Problem Let G Be a Lie Algebra And E Amentioning

confidence: 99%

Extending structures for Lie algebras

Agore

Militaru

2013

Monatsh Math

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Abstract. Let g be a Lie algebra, E a vector space containing g as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on E such that g is a Lie subalgebra of E. A general product, called the unified product, is introduced as a tool for our approach. Let V be a complement of g in E: the unified product g ♮ V is associated to a system (⊳, ⊲, f, {−, −}) consisting of two actions ⊳ and ⊲, a generalized cocycle f and a twisted Jacobi bracket {−, −} on V . There exists a Lie algebra structure [−, −] on E containing g as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras (E, [−, −]) ∼ = g ♮ V . All such Lie algebra structures on E are classified by two cohomological type objects which are explicitly constructed. The first one H 2 g (V, g) will classify all Lie algebra structures on E up to an isomorphism that stabilizes g while the second object H 2 (V, g) provides the classification from the view point of the extension problem. Several examples that compute both classifying objects H 2 g (V, g) and H 2 (V, g) are worked out in detail in the case of flag extending structures.

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“…This is what we do in Section 2. For further reference on the extension problem for Lie algebras, in the abelian or non-abelian case, we refer to [3], [4], [9], [11], [13], [24].…”

Section: Extending Structures Problem Let G Be a Lie Algebra And E Amentioning

confidence: 99%

Extending structures for Lie algebras

Agore

Militaru

2013

Monatsh Math

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“…Note that (11) means that l x ∈ Der L (h), and (12) means that r x ∈ Der R (h). (17). We denote by Z 2 (g, h) the set of non-abelian 2-cocycles.…”

Section: Classification Of Non-abelian Extensions Of Leibniz Algebrasmentioning

confidence: 99%

On non-abelian extensions of Leibniz algebras

Liu

Sheng

Wang

2017

Communications in Algebra

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In this paper, first we classify non-abelian extensions of Leibniz algebras by the second non-abelian cohomology. Then, we construct Leibniz 2-algebras using derivations of Leibniz algebras, and show that under a condition on the center, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. At last, we give a description of non-abelian extensions in terms of Maurer-Cartan elements in a differential graded Lie algebra.

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“…bilinear maps f : L × L → g satisfying the compatibility condition (CS3). Two such cocycles f and f ′ are (local) cohomologous and we denote this by f ≈ l,a f ′ if there exists a linear map r : L → g such that (20) holds. Now, ≈ l,a is an equivalent relation on the set ZL 2 (⊳, ⊲) (L, g 0 ) of all 2-cocycles and the quotient set ZL 2 (⊳, ⊲) (L, g 0 )/ ≈ l,a is just the second cohomological group [25, Proposition 1.9] and is denoted by HL 2 (⊳, ⊲) (L, g 0 ).…”

Section: The Global Extension Problem For Leibniz Algebrasmentioning

confidence: 99%

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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A cohomological characterization of Leibniz central extensions of Lie algebras

Cited by 30 publications

References 22 publications

Extending structures for Lie algebras

Extending structures for Lie algebras

On non-abelian extensions of Leibniz algebras

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

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