Manin triples of 3-Lie algebras induced by involutive derivations

Abstract

For any $n$-dimensional 3-Lie algebra $A$ over a field of characteristic zero with an involutive derivation $D$, we investigate the structure of the 3-Lie algebra $B_1=A{⋉}_{a{d}^{\ast }}{A}^{\ast }$ associated with the coadjoint representation $(A^{\ast },a{d}^{\ast })$. We then discuss the structure of the dual 3-Lie algebra $B_2$ of the local cocycle 3-Lie bialgebra $(A{⋉}_{a{d}^{\ast }}{A}^{\ast },\Delta )$. By means of the involutive derivation $D$, we construct the $4n$-dimensional Manin triple $(B_1\oplus B_2,$ $[\cdot ,\cdot ,\cdot {]}_1,$ $[\cdot ,\cdot ,\cdot {]}_2,$ $B_1,B_2)$ of 3-Lie algebras, and provide concrete multiplication in a special basis ${\Pi }_1\cup {\Pi }_2$. We also construct a sixteen dimensional Manin triple $(B,[\cdot ,\cdot ,\cdot ])$ with $\dim B^1=12$ using an involutive derivation on a four dimensional 3-Lie algebra $A$ with $\dim A^1=2$.