Geodesic
Manin Triples of 3-Lie Algebras Induced by Involutive Derivations
Journal of Lie theory, Tome 30 (2020) no. 1, pp. 239-257
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\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.
Classification : 16T10, 16T25, 17A30, 17B62
Mots-clés : 3-Lie algebra, involutive derivation, semi-direct product 3-Lie algebra, Manin triple, 3-Lie bialgebra 
@article{JLT_2020_30_1_JLT_2020_30_1_a13,
     author = {S. Hou and R. Bai and Y. Sheng},
     title = {Manin {Triples} of {3-Lie} {Algebras} {Induced} by {Involutive} {Derivations}},
     journal = {Journal of Lie theory},
     pages = {239--257},
     year = {2020},
     volume = {30},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2020_30_1_JLT_2020_30_1_a13/}
}
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S. Hou; R. Bai; Y. Sheng. Manin Triples of 3-Lie Algebras Induced by Involutive Derivations. Journal of Lie theory, Tome 30 (2020) no. 1, pp. 239-257. http://geodesic.mathdoc.fr/item/JLT_2020_30_1_JLT_2020_30_1_a13/