Symplectic Structures On Metric 3-Lie Algebras

3-Lie algebra is a branch of mathematics, whose applicability is very strong.Especially, metric 3-Lie algebras are widely used in many fields of mathematics and mathematical physics. Based on the metric 3-Lie algebras, we study the symplectic structures of metric 3-Lie algebras. The main work of the paper consists of three sections:(i) The definition and properties of sympletic structure of 3-Lie algebras are given. 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 ∈ DerB(A).Form arbitrary 3-Lie algebra L, we can construct metric sympectic 3-Lie algebras.(ii) We studied the structures of T?θ-extension of 3-Lie algebras, and the necessary and su?cient condition of the symplectic structure of nilpotent metric 3-Lie algebras is provided. We also proved that every metric symplectic 3-Lie algebra(?A,?B, ?ω) is a T? extension of a metric symplectic 3-Lie algebra(A, B, ω). And then we construct a double extension of a metric symplectic 3-Lie algebra by means of a special derivation.(iii) We give the double extension of a 3-Lie algebra by a special derivation, an discuss the symplectic structure of it.