Cohomologies of a Lie algebra with a derivation and applications

Tang, Rong; Frégier, Yaël; Sheng, Yunhe

doi:10.1016/j.jalgebra.2019.06.007

Cited by 39 publications

(30 citation statements)

References 27 publications

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“…Note that derivations are 1-cocycles in the cohomology complex of g with coefficients in g. We call a pair (g, φ g ) of a Leibniz algebra g and a derivation φ g , a LeibDer pair. When the Leibniz algebra bracket is skew-symmetric, one gets the notion of LieDer pair [26]. Thus, LeibDer pairs are non-skewsymmetric analog of LieDer pairs.…”

Section: Leibder Pairsmentioning

confidence: 99%

“…In [14,22] the authors study algebras with derivations from operadic point of view. Recently, Lie algebras with derivations (called LieDer pairs) are studied from cohomological point of view [26] and extensions, deformations of LieDer pairs are considered. The results of [26] has been extended to associative algebras with derivations (called AssDer pairs) in [13].…”

Section: Introductionmentioning

confidence: 99%

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Leibniz algebras with derivations

Das¹

2020

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In this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions and abelian extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz bracket and the distinguished derivation.It is governed by the cohomology of LeibDer pair with coefficients in itself. Finally, we consider homotopy derivations on sh Leibniz algebras and 2-derivations on Leibniz 2-algebras. The category of 2-term sh Leibniz algebras with homotopy derivations is equivalent to the category of Leibniz 2-algebras with 2-derivations. Contents 1. Introduction 1 2. Leibniz algebras and their cohomology 3 3. LeibDer pairs 4 3.1. Universal enveloping AssDer pair 6 3.2. Cohomology 7 4. Extensions of LeibDer pairs 8 4.1. Central extensions 8 4.2. Extensions of a pair of derivations 9 4.3. Abelian extensions 11 5. Deformations of LeibDer pairs 12 5.1. Rigidity 13 5.2. Finite order deformations and their extensions 13 6. Homotopy derivations on sh Leibniz algebras 15 7. Categorification of LeibDer pairs 18 8. Conclusions 20 References 20

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Section: Leibder Pairsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Leibniz algebras with derivations

Das¹

2020

Preprint

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“…In [24] the authors call the pair (g, φ g ) of a Lie algebra and a derivation by a LieDer pair. Here we introduce higher derivations on a Lie algebra similar to the case of associative algebras.…”

Section: Examplementioning

confidence: 99%

“…Algebras with derivations are studied from operadic point of view in [19]. Recently, Lie algebras with derivations (called LieDer pairs) are studied from cohomological perspectives and extension, deformation of LieDer pairs are considered [24]. These results has been extended to associative algebras and Leibniz algebras with derivations (AssDer pairs and LeibDer pairs) in [7,8].…”

Section: Introductionmentioning

confidence: 99%

Extensions and deformations of algebras with higher derivations

Das¹

2020

Preprint

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Higher derivations on an associative algebra generalizes higher order derivatives. We call a tuple consisting of an algebra and a higher derivation on it by an AssHDer pair. We define a cohomology for AssHDer pairs with coefficients in a representation. Next, we study central extensions, abelian extensions of AssHDer pairs and relate them to the second cohomology group with appropriate coefficients. Deformations of AssHDer pairs are also considered which are governed by the cohomology with self coefficient.

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“…In [16,23], the authors study algebras with derivations from the operadic point of view. Recently, the authors in [30] study the cohomology, extensions and deformations of Lie algebras with derivations (called LieDer pairs). The results of [30] have been extended to associative algebras with derivations (called AssDer pairs) and Leibniz algebras with derivations (called LeibDer pairs) in [14,15].…”

Section: Introductionmentioning

confidence: 99%

Cohomologies of 3-Lie algebras with derivations

Xu,

Liu

2021

Preprint

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In this paper, we consider a 3-Lie algebra with a derivation (called a 3-LieDer pair). We define cohomology for a 3-LieDer pair with coefficients in a representation. We use this cohomology to study deformations and abelian extensions of 3-LieDer pairs. We give the notion of a 3-Lie2Der pair, which can be viewed as the categorification of a 3-LieDer pair. We show that skeletal 3-Lie2Der pairs are classified by triples given by 3-LieDer pairs, representations and 3cocycles. We define crossed modules of 3-LieDer pairs and show that there exists a one-to-one correspondence between strict 3-Lie2Der pairs and crossed modules of 3-LieDer pairs.

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Cohomologies of a Lie algebra with a derivation and applications

Cited by 39 publications

References 27 publications

Leibniz algebras with derivations

Leibniz algebras with derivations

Extensions and deformations of algebras with higher derivations

Cohomologies of 3-Lie algebras with derivations

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