BAI Ruipu, Li Qiyong and Cheng Rong
The realization of n-Lie algebras is very important in the study of the structure of n-Lie algebras for n ≥ 3. This paper considers the realizations of 3-Lie algebras by cubic matrices. First, the trace function tr1 of cubic matrices is defined, and then the 3-ary Lie multiplication [, , ]tr1 on the vector space Ω spanned by cubic matrices is constructed, and the structure of the 3-Lie algebra (Ω, [, , ]tr1 ) is investigated. It is proved that (Ω, [, , ]tr1 ) is a decomposable 3-Lie algebra, and there does not exist any metric on it.