Structures Of3-Lie Coalgebras

N-Lie algebras are generalizations of Lie algebras, 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 important models are constructed on structures of3-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 of3-Lie coalgebras over a field of characteristic0. The concept of3-Lie coalgebras is provided and the structure is studied. The relations between the3-Lie coalgebras and3-Lie algebras are also studied. At the last of the paper, classifications of3-dimensional Lie coalgebras over an algebraically closed field of characteristic0is studied.The paper consists of five sections. Section1introduces the back ground and de-velopment of n-Lie algebras. Section2provides some definitions and some basic results which are used in the paper. Section3defines the concept of3-Lie coalgebra and studies structures. Section4studies the relations between3-Lie coalgebras and3-Lie algebras. The last section gives the summarization of the paper and the prospects of the study.