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تسجيل مستخدم جديدSymplectic structures on $3$-Lie algebras
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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 $Din Der_B(A)$, 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.
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