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

Abstract

In this paper, we give 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 dimensions 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 eight-dimensional pseudo-metric 3-Lie algebras.

Keywords:

• Bialgebra
• 3-Lie algebra
• 3-Lie bialgebra
• classical Yang–Baxter equation
• Manin triple

AMS Subject Classifications:

• 15A75
• 16T10
• 16T25
• 17A30
• 17B62
• 81T30

Acknowledgements

The authors thank Jim Stasheff for encouragement and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Natural Science Foundation of China [grant number 11371178], [grant number 11425104].