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

• DOI: 10.4310/atmp.2019.v23.n1.a2
• Publication Date: 2019-09-27T17:57:17Z
• Authors: Chengming Bai; Li Guo; Yunhe Sheng
• Keywords: Bialgebra; Simple Lie group; Yang–Baxter equation

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