Biderivations of finite-dimensional complex simple Lie algebras

Tang, Xiaomin

doi:10.1080/03081087.2017.1295433

Cited by 41 publications

(13 citation statements)

References 19 publications

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“…A mapping φ of a Lie algebra g is called commuting if [φ(x), x] = 0 for any x in g (see [2]). An important application of linear commuting mappings is to construct biderivations (for example, see [2,6,20,21]), as shown in the following lemma. Lemma 1.2.…”

Section: Introductionmentioning

confidence: 99%

Biderivations and linear commuting mappings of $\mathbb{N}$-graded Lie algebras of maximal class

Yang¹,

Tang²,

Chen³

2021

Preprint

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Combined with Fialowski's classification for algebras of maximal class and using the first cohomology of these Lie algebras with coefficients in the adjoint module, in this paper we determine the skew-symmetric biderivation spaces for all algebras of maximal class with 1-dimensional homogeneous components m 0 , L 1 and m 2 as the given gradings, up to isomorphism. Different from the usual conclusions, we obtain that a non-simple Lie algebra L 1 only has inner biderivations. As an application, we characterize linear commuting mappings on these algebras.

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Section: Introductionmentioning

confidence: 99%

Biderivations and linear commuting mappings of $\mathbb{N}$-graded Lie algebras of maximal class

Yang¹,

Tang²,

Chen³

2021

Preprint

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“…In particular, biderivations are closely related to the theory of commuting linear maps that has a long and rich history, and we refer to the survey [4] for the development of commuting maps and their applications. It is worth mentioning that Brešar and Zhao considered a general but simple approach for describing biderivations and commuting linear maps on a Lie algebra L that having their ranges in an L-module [6], which covered most of the results in [11,14,22,29,31], and inspires us to generalize their method to Hom-Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

Biderivations and commuting linear maps on Hom-Lie algebras

Sun,

Ma,

Chen

2020

Preprint

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The purpose of this paper is to determine skew-symmetric biderivations Bider s (L,V ) and commuting linear maps Com(L,V ) on a Hom-Lie algebra (L, α) having their ranges in an (L, α)-module (V, ρ, β ), which are both closely related to Cent(L,V ), the centroid of (V, ρ, β ). Specifically, under appropriate assumptions, every δ ∈ Bider s (L,V ) is of the form δ (x, y) = β −1 γ([x, y]) for some γ ∈ Cent(L,V ), and Com(L,V ) coincides with Cent(L,V ). Besides, we give the algorithm for describing Bider s (L,V ) and Com(L,V ) respectively, and provide several examples.

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“…In [7], the authors determined all the skewsymmetric biderivations of W(a;b) and found that there exist non-inner biderivations. In [12,13], the authors characterized the biderivations without the anti-symmetric condition of the finite dimensional complex simple Lie algebra and some W-algebras, meanwhile, presented some classes of non-inner biderivations. In 2015, Xu and Wang generalized the notion of biderivations of Lie algebra to the super case in [16], and the authors mainly discussed some properties of super-biderivations on Heisenberg superalgebras.…”

Section: Introductionmentioning

confidence: 99%