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.