2019
DOI: 10.1016/j.jalgebra.2019.06.007
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Cohomologies of a Lie algebra with a derivation and applications

Abstract: The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology gr… Show more

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Cited by 40 publications
(30 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…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%
“…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%