Some Properties Of 3-Lie Algebra

In recent years,the study of lie algebra is getting deeper and deeper.A3-lie algebra is a natural generalization of lie algebra,which is closely related to many fields in mathematics and physics.In this paper,we first study the multiplicative tables of 3-lie algebra of 7 dimensional on the closed field F with characteristic 0 and with derivation algebras of dimension 1,and then obtain the nilpotent and solvable properties of 3-lie algebra under corresponding multiplication tables.Then,in this paper,we give the structure matrix of 7dimensional 3-lie algebra which has an ideal of 6 dimensional linear 3-lie algebra on an algebraic closed field F with characteristic 0.Then we study the structure of the non-nilpotent 7 dimensional 3-lie algebra with 6 dimensional linear3-lie algebra as a hypo-nilpotent ideal,and obtain that the 7 dimensional 3-lie algebra is solvable.Finally,the existence of the corresponding metric structure of low dimensional 3-lie algebra is proved.