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

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n1.a2

Authors

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)

Abstract

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.

Published 27 September 2019