The Classification Of3-Lie Bialgebras

We know that in the structural researching, the classification study of an algebrasystem is basic and also difcult. Since there are complete structural classifications andcomplete classifications of irreducible representations on the finite dimensional semisim-ple Lie algebra over the complex field, the structural theory of the finite dimensionalsemisimple Lie algebra is perfect. In this paper we concentrate our main attension tothe classification of3-Lie bialgebras over an algebrically closed field of characteristic0.The classification of3-dimensional3-Lie bialgebras and4-dimensional3-Lie bialgebrasare studied. It is proved that there are only six classes and thirteen classes of non-isomorphic3-dimensional3-Lie bialgebras and4-dimensional3-Lie bialgebras (with thedual dimension less than2), respectively. And the multiplication tables are provided.The paper consists of four sections. Section1introduces the back ground anddevelopment of n-Lie algebras and Lie algebras. Section2gives some definitions andproves some basic results which are used in the paper. Section3studies the classificationof3-dimensional3-Lie bialgebras. Section4studies the classification of4-dimensional3-Lie bialgebras. The last section is the summarization of the paper and gives the prospectsof the study.