Symplectic structures on $3$-Lie algebras

Abstract

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 $D\in De{r}_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(\tilde{A},\tilde{B},\tilde{\omega })$ is a $T_{\theta }^{\ast }$-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.

arXiv admin note: text overlap with arXiv:math/0603066 by other authors

References in corpus (1)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes

Cited by in corpus (2)

• 3-Lie-Rinehart Algebras
• Hom 3-Lie-Rinehart Algebras