Structures Of 5-dimensional 3-lie Algebras

N-Lie algebra is a generaliztion of Lie algebra, which is an algebraic system with an n-ary multilinear operation, and it has wide applications in geometries, mechanics and string theories. In physics, some impotant models are constructed on the structures of metric 3-Lie algebras. But the structural theory of n-Lie algebras is imperfectness, which obstructs the application of n-Lie algebras. The paper mainly concerns structures of 5-dimensional 3-Lie algebras over a complete field of characteristic 2. It proved that the 5-dimensional 3-Lie algebras are 2-solvable in the cases of the dimension of the derived algebra is not more than 2, and that are 3-solvable but not 2-solvable in the cases of the dimension of the derived algebra is 3. It also proved that there exists only one calss of 2-semisimple 5-dimensional 3-Lie algebras over a complete field of characteristic 2. The paper also discussed the solvability and nilpotency of 5-dimensional 3-Lie algebras and Cartan subalgebras.The paper consists of five sections. Section 1 introduced the back ground and development of n-Lie algebras. Section 2 recalled some definitions and results which are used in the paper. Section 3 studied the solvibility and nilpotency of 5-dimensional 3-Lie algebras. Section 4 discussed the Cartan subalgebras of every class. The last section is the summarizing of the paper.