Constructing 3-Lie algebras

Abstract: 3-Lie algebras are constructed by Lie algebras, derivations and linear functions, associative commutative algebras, whose involutions and derivations. Then the 3-Lie algebras are obtained from group algebras $F[G]$. An infinite dimensional simple 3-Lie algebra $(A, [,,]{\omega, \delta_0})$ and a non-simple 3-Lie algebra $(A, [,,]{\omega_1, \delta})$ are constructed by Laurent polynomials $A=F[t, t^{-1}]$ and its involutions $\omega$ and $\omega_1$ and derivations $\delta$ and $\delta_0$. At last of the paper, we summarize the methods of constructing $n$-Lie algebras for $n\geq 3$ and provide a problem.