Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 1

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

Pages: 27 – 74



Chengming Bai (Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, China)

Li Guo (Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi, China; and Department of Mathematics & Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)

Yunhe Sheng (Department of Mathematics, Jilin University, Changchun, Jilin, China)


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 $\mathcal{O}$-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.

This research is supported by NSFC (11471139, 11425104, 11771190) and NSF of Jilin Province (20170101050JC). C. Baiis also supported by the Fundamental Research Funds for the Central Universities and Nankai ZhiDe Foundation.

Published 27 September 2019