n-Lie bialgebras

Abstract

We define n-Lie coalgebras with rank r and discuss their structures. We also introduce n-Lie bialgebras and investigate their structures. A triple is an n-Lie bialgebra if and only if is a conformal 1-cocycle on the n-Lie algebra L associated to L-modules , . Furthermore we study two-dimensional extensions of finite dimensional n-Lie bialgebras, and construct an -dimensional -Lie bialgebra associated to an m dimensional n-Lie bialgebra . Finally we discuss the bialgebra structure on the finite dimensional simple n-Lie algebra , and prove that the only bialgebra structures on the simple n-Lie algebra are of rank zero and rank two.

Keywords:

• n-Lie algebra
• n-Lie bialgebra
• n-Lie coalgebra
• two-dimensional extension

AMS Subject Classifications:

• 17B05
• 17D99

Acknowledgements

The authors would like to thank Professor Chengming Bai for many valuable suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported in part by the Natural Science Foundation [grant number 11371245]; Natural Science Foundation of Hebei Province [grant number A2014201006].