Cohomology and deformations of Relative Rota-Baxter operators on Lie-Yamaguti algebras

Zhao, Jia; Qiao, Yu

doi:10.48550/arxiv.2204.04872

Cited by 6 publications

(10 citation statements)

References 29 publications

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“…the sub-adjecent Lie-Yamaguti algebra given by ( 23) and (24). Recall that in [30], we constructed a representation of V T on g. Lemma 5.4. Define linear maps ̺ T : V → gl(g) and ̟ T :…”

Section: Maurer-cartan Operators On Twilled Lie-yamaguti Algebrasmentioning

confidence: 99%

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Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

Zhao¹,

Qiao²

2022

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A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory of LieYRep pairs, and characterize their linear deformations by the first cohomology group. Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs and investigate their properties. As applications, we show that a relative Rota-Baxter-Nijenhuis structure gives rise to a Maurer-Cartan operator on a twilled Lie-Yamaguti algebra under certain condition, and obtain the equivalence between r-matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures on Lie-Yamaguti algebras.

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Section: Maurer-cartan Operators On Twilled Lie-yamaguti Algebrasmentioning

confidence: 99%

“…The first author and Sheng focused on deformations, Nijenhuis operators, and relative Rota-Baxter operators on Lie-Yamaguti algebras in [22,23]. Recently, we studied cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras ( [30]).…”

Section: Introductionmentioning

confidence: 99%

Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

Zhao¹,

Qiao²

2022

Preprint

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“…Furthermore, Benito, Bremmer, and Madariaga examined orthogonal Lie-Yamaguti algebras in [12]. Recently, we studied cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras [31], relative Rota-Baxter-Nijenhuis structures on a Lie-Yamaguti algebra with a representation [32], and bialgebra theory of Lie-Yamaguti algebras [33].…”

Section: Introductionmentioning

confidence: 99%

Para-Kähler and pseudo-Kähler structures on Lie-Yamaguti algebras

Zhao¹,

Feng²,

Qiao³

2023

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For a pre-Lie-Yamaguti algebra A, by using its sub-adjacent Lie-Yamaguti algebra A c , we are able to construct a semidirect product Lie-Yamaguti algebra via a representation of A c . The investigation of such semidirect Lie-Yamaguti algebras leads us to the notions of para-Kähler structures and pseudo-Kähler structures on Lie-Yamaguti algebras, and also gives the definition of complex product structures on Lie-Yamaguti algebras which was first introduced in [25]. Furthermore, a Levi-Civita product with respect to a pseudo-Riemannian Lie-Yamaguti algebra is introduced and we explore its relation with pre-Lie-Yamaguti algebras.

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“…Sheng and the first author focused on linear deformations, product structures and complex structures on Lie-Yamaguti algebras in [21] and later, relative Rota-Baxter operators and pre-Lie-Yamaguti algebras were introduced in [22]. Besides, we studied cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras in [30].…”

Section: Introductionmentioning

confidence: 99%

“…Note that when ρ 1 = ad * , µ 1 = −R * τ and ρ 2 = ad * , µ 2 = −R * τ, Eqs. (27)-(30) and Eqs. (32)-(34) are equivalent to Conditions (45)-(51) respectively, Eq.…”

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confidence: 99%

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

Zhao¹,

Qiao²

2022

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In this paper, we develop the bialgebra theory for Lie-Yamaguti algebras. For this purpose, we exploit two types of compatibility conditions: local cocycle condition and double construction. We define the classical Yang-Baxter equation in Lie-Yamaguti algebras and show that a solution to the classical Yang-Baxter equation corresponds to a relative Rota-Baxter operator with respect to the coadjoint representation. Furthermore, we generalize some results by Bai in [1] and Semonov-Tian-Shansky in [19] to the context of Lie-Yamaguti algebras. Then we introduce the notion of matched pairs of Lie-Yamaguti algebras, which leads us to the concept of double construction Lie-Yamaguti bialgebras following the Manin triple approach to Lie bialgebras. We prove that matched pairs, Manin triples of Lie-Yamaguti algebras, and double construction Lie-Yamaguti bialgebras are equivalent. Finally, we clarify that a local cocycle condition is a special case of a double construction for Lie-Yamaguti bialgebras. Contents 1. Introduction 1 2. Preliminaries 4 3. Relative Rota-Baxter operators, the classical Yang-Baxter equation, and local cocycle Lie-Yamaguti bialgebras 8 4. Manin triples, matched pairs, and double construction Lie-Yamaguti bialgebras 18 References 25

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Cohomology and deformations of Relative Rota-Baxter operators on Lie-Yamaguti algebras

Cited by 6 publications

References 29 publications

Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

Para-Kähler and pseudo-Kähler structures on Lie-Yamaguti algebras

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

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