# 3-Lie bialgebras and 3-Lie classical Yang-Baxter equations in low dimensions

## TL;DR

This paper explores low-dimensional examples of 3-Lie bialgebras, providing explicit formulas for solutions to the 3-Lie classical Yang-Baxter equation and classifying related structures in dimensions 3 and 4.

## Contribution

It introduces practical formulas for skew-symmetric solutions of the 3-Lie CYBE and classifies 3-Lie bialgebras and Manin triples in low dimensions.

## Key findings

- Explicit solutions to 3-Lie CYBE in dimensions 3 and 4
- Classification of 3-Lie bialgebras in low dimensions
- Construction of 8-dimensional pseudo-metric 3-Lie algebras

## Abstract

In this paper, we give some low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang-Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimension 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4 and then give the corresponding 8-dimensional pseudo-metric 3-Lie algebras.