TY - JOUR

T1 - Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras

AU - Bai, Chengming

AU - Guo, Li

AU - Sheng, Yunhe

N1 - Funding Information:
Acknowledgements. This research is supported by NSFC (11471139, 11425104, 11771190) and NSF of Jilin Province (20170101050JC). C. Bai is also supported by the Fundamental Research Funds for the Central Universities and Nankai ZhiDe Foundation.
Publisher Copyright:
© 2019 International Press of Boston, Inc.

PY - 2019

Y1 - 2019

N2 - This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are thus called the local cocycle 3-Lie bialgebra and double construction 3-Lie bialgebra. They are two extensions of the wellknown Lie bialgebra in the context of 3-Lie algebras. The local cocycle 3-Lie bialgebra extends the connection between Lie bialgebras and the classical Yang-Baxter equation. Its relationship with a ternary variation of the classical Yang-Baxter equation, called the 3-Lie classical Yang-Baxter equation, a ternary O-operator and a 3-pre-Lie algebra is established. In particular, solutions of the 3-Lie classical Yang-Baxter equation give (coboundary) local cocycle 3-Lie bialgebras, whereas, 3-pre-Lie algebras give rise to solutions of the 3-Lie classical Yang-Baxter equation. The double construction 3-Lie bialgebra is also introduced to extend to the 3-Lie algebra context the connection between Lie bialgebras and double constructions of Lie algebras. Their related Manin triples give a natural construction of pseudo-metric 3-Lie algebras with neutral signature. The double construction 3-Lie bialgebra can be regarded as a special class of the local cocycle 3-Lie bialgebra. Explicit examples of double construction 3-Lie bialgebras are provided.

AB - This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are thus called the local cocycle 3-Lie bialgebra and double construction 3-Lie bialgebra. They are two extensions of the wellknown Lie bialgebra in the context of 3-Lie algebras. The local cocycle 3-Lie bialgebra extends the connection between Lie bialgebras and the classical Yang-Baxter equation. Its relationship with a ternary variation of the classical Yang-Baxter equation, called the 3-Lie classical Yang-Baxter equation, a ternary O-operator and a 3-pre-Lie algebra is established. In particular, solutions of the 3-Lie classical Yang-Baxter equation give (coboundary) local cocycle 3-Lie bialgebras, whereas, 3-pre-Lie algebras give rise to solutions of the 3-Lie classical Yang-Baxter equation. The double construction 3-Lie bialgebra is also introduced to extend to the 3-Lie algebra context the connection between Lie bialgebras and double constructions of Lie algebras. Their related Manin triples give a natural construction of pseudo-metric 3-Lie algebras with neutral signature. The double construction 3-Lie bialgebra can be regarded as a special class of the local cocycle 3-Lie bialgebra. Explicit examples of double construction 3-Lie bialgebras are provided.

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U2 - 10.4310/ATMP.2019.v23.n1.a2

DO - 10.4310/ATMP.2019.v23.n1.a2

M3 - Article

AN - SCOPUS:85077587686

VL - 23

SP - 27

EP - 74

JO - Advances in Theoretical and Mathematical Physics

JF - Advances in Theoretical and Mathematical Physics

SN - 1095-0761

IS - 1

ER -