Structures Of Low Dimensional3-Lie Bialgebras

Lie bialgebra is a Lie algebra wcich is compatible with a structure of Lie algebra on the dual space. Lie bialgebra has wide applications in mathematical physics. So the bialgebric structure of multiple Lie algebra is studied in recent years. The concepts of3-Lie coalgebra and3-Lie bialgebra are provided, but the structural investigation is basic. Therefore, it is natural to study the structures of low dimensional3-Lie coalgebras and low dimensional3-Lie bialgebras. In this paper we concentrate our main attension to the structure of3-Lie bialgebras over an algebrically closed field of characteristic0. Basing on the classification of low dimensional3-Lie algebras and low dimensional3-Lie coalgebras, we study the classification of4-dimensional3-Lie bialgebras (L,μbi,Δd) and (L,μbi,Δe), where (L,μbi), i=1,2are4-dimensional3-Lie algebras with dim L1=1, and (L, Δd), and (L,Δe) are4-dimensional3-Lie coalgebras with the dual3-Lie algebra (L*,Δd*) and (L*,Δe*) of dim(L*)1=3,4. It is proved that there are only three classes non-equalent3-Lie bialgebras (L, μbi,Δe). And the concete multiplications are provided.The paper consists of four sections. Section1introduces the back ground and development of n-Lie algebras and3-Lie bialgebras. Section2gives some definitions and proves some basic results which are used in the paper. Section3studies classifications of4-dimensional3-Lie bialgebras (L,μbi,Δd) and compatibility(L,μbi,Δe). The last section is the summarization of the paper and gives the prospects of the study.