Journal of Lie Theory 30 (2020), No. 1, 239--257
Copyright Heldermann Verlag 2020
Manin Triples of 3-Lie Algebras Induced by Involutive Derivations
Shuai Hou
Dept. of Mathematics, Jilin University, Changchun 130012, P. R. China
and: College of Mathematics and Information Science, Hebei University, Baoding 071002, P. R. China
hshuaisun@163.com
Ruipu Bai
College of Mathematics and Information Science, Hebei University, Baoding 071002, P. R. China
bairuipu@hbu.edu.cn
Yunhe Sheng
Dept. of Mathematics, Jilin University, Changchun 130012, P. R. China
shengyh@jlu.edu.cn
[Abstract-pdf]
\newcommand{\ad}{\mathrm{ad}} Any involutive derivation $D$ on a 3-Lie algebra $A$ induces a local cocycle 3-Lie bialgebra $(A\ltimes_{\ad^*} A^*, \Delta)$. We give precise formulas of the 3-Lie algebra $((A\oplus A^*)^*, \Delta^*)$ and show that the local cocycle 3-Lie bialgebra $(A\ltimes_{\ad^*} A^*, \Delta)$ induced by the involutive derivation $D$ gives rise to a Manin triple of 3-Lie algebras. We give examples of $12$-dimensional and $16$-dimensional Manin triples using involutive derivations on certain $3$-dimensional and $4$-dimensional $3$-Lie algebras.
Keywords: 3-Lie algebra, involutive derivation, semi-direct product 3-Lie algebra, Manin triple, 3-Lie bialgebra.
MSC: 16T10, 16T25, 17A30, 17B62.
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