Symplectic structures on $3$-Lie algebras

The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $D\in Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(\tilde{A}, \tilde{B}, \tilde{\omega})$ is a $T^*_{\theta}$-extension of a metric symplectic $3$-Lie algebra $(A, B, \omega)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.