Bialgebras, the classical Yang–Baxter equation and Manin triples for 3-Lie algebras

Bai, Chuanzhi; Guo, Li; Sheng, Yunhe

doi:10.4310/atmp.2019.v23.n1.a2

Cited by 53 publications

(58 citation statements)

References 28 publications

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“…But a perfect product structure E on (g, [·, ·, ·] g ) ensures [g + , g + , g − ] g ⊂ g − and [g − , g − , g + ] g ⊂ g + . Note that this is exactly the condition required in the definition of a matched pair of 3-Lie algebras [11]. Thus, E is a perfect product structure if and only if (g + , g − ) is a matched pair of 3-Lie algebras.…”

Section: Definition 516 (Integrability Condition Iv)mentioning

confidence: 85%

“…This is also a new phenomenon. Note that the decomposition that a perfect product structure gives is exactly the condition required in the definition of a matched pair of 3-Lie algebras [11]. Thus, this kind of product structure will be frequently used in our studies.…”

Section: Introductionmentioning

confidence: 94%

“…Lemma 2.5. ( [11]) Let (V, ρ) be a representation of a 3-Lie algebra (g, [·, ·, ·] g ). Then (V * , ρ * ) is a representation of the 3-Lie algebra (g, [·, ·, ·] g ), which is called the dual representation.…”

Section: Preliminariesmentioning

confidence: 99%

“…The notions of a matched pair of 3-Lie algebras and a Manin triple of 3-Lie algebras were introduced in[11]. By(29), we obtain that (A c , A * c ; L * , L * ) is a matched pair of 3-Lie algebras and the phase space is exactly the double of this matched pair.…”

mentioning

confidence: 99%

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Symplectic, product and complex structures on 3-Lie algebras

Sheng

Tang

2018

Journal of Algebra

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In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to O-operators, Rota-Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-Kähler structure and a pseudo-Kähler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.

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Section: Definition 516 (Integrability Condition Iv)mentioning

confidence: 85%

Section: Introductionmentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

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Symplectic, product and complex structures on 3-Lie algebras

Sheng

Tang

2018

Journal of Algebra

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“…In paper [1], authors studied local cocycle 3-Lie bialgebras (A, µ, ∆), and defined 3-Lie classical Yang-Baxter equation (…”

Section: Introductionmentioning

confidence: 99%

n-Lie bialgebras

Bai

Guo

Lin

et al. 2017

Linear and Multilinear Algebra

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The n-Lie bialgebras are studied. In Section 2, the n-Lie coalgebra with rank r is defined, and the structure of it is discussed. In Section 3, the n-Lie bialgebra is introduced. A triple (L, µ, ∆) is an n-Lie bialgebra if and only if ∆ is a conformal 1-cocycle on the n-Lie algebra L associated to L-modules (L ⊗n , ρ µ s ), 1 ≤ s ≤ n, and the structure of n-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional n-Lie bialgebras are studied. For an m dimensional n-Lie bialgebra (L, µ, ∆), and an ad µ -invariant symmetric bilinear form on L, the m + 2 dimensional (n + 1)-Lie bialgebra is constructed. In the last section, the bialgebra structure on the finite dimensional simple n-Lie algebra A n is discussed. It is proved that only bialgebra structures on the simple n-Lie algebra A n are rank zero, and rank two.

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Symplectic, Product and Complex Structures on 3‐Lie Algebras

Sheng

Tang

2021

Algebra and Applications 1

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Bialgebras, the classical Yang–Baxter equation and Manin triples for 3-Lie algebras

Cited by 53 publications

References 28 publications

Symplectic, product and complex structures on 3-Lie algebras

Symplectic, product and complex structures on 3-Lie algebras

n-Lie bialgebras

Symplectic, Product and Complex Structures on 3‐Lie Algebras

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