Invariant Bilinear Forms On 3-Lie Algebras

N-Lie algebra is a generaliztion of Lie algebra, which is an algebraic system with an n-ary multilinear operation. The n-Lie algebra has its wide applications in geometries, mechanics and string theories. We know that some impotant M2-models are constructed on structures of metric 3-Lie algebras. But the structure theory of metric n-Lie algebras is im-perfectness. Therefore, the study of symmetric invariant bilinear forms on 3-Lie algebras is necessary. The paper studies the structure of symmetric invariant bilinear forms on 3-Lie algebras, and gives the concrete expres-sion of every symmetric invariant bilinear form on the 5-dimensional 3-Lie algebras over a complete field with characteristic 2. At last, it proves that there not exist the non-degenerate symmetric invariant bilinear forms on the 5 diemnsional 3-Lie algebras with 1,2,3 dimensional derived algebras. And the 5 diemnsional 3-Lie algebras with 4 dimensional derived algebras are metric 3-Lie algebras.The paper consists of three sections. The back ground and develop-ment of n-Lie algebras are introduced in section 1. Section 2 gives some definitions and results of 3-Lie algebras which are used in the paper. Sec-tion 3 studies the structure of symmetric invariant bilinear forms on 3-Lie algebras.