Rota-Baxter Operators Of The Simple 3-Lie Algebra

In recent years, the Rota-Baxter operator has been well applied in many areas of mathematics and physics. For example, as a solution of the classical condition of Yang-Baxter equation it in the form of operators appeared in the application of Lie algebra, and it plays an important role in the integration system. Currently the Rota-Baxter operator was found to have important applications in quantum theory.Moreover, Rota- Baxter Lie algebra is closely related to the complex adjoint algebras of left-symmetry algebra. So it is natural to study the Rota-Baxtor n-Lie algebra.The main work of this paper can be summarized as four aspects:(i)The general conditions of Rota-Baxter operator of the simple 3- Lie algebra are given.(ii)A method for the Rota-Baxter operator of the simple 3- Lie algebra classification when its weight is 0 and rank is 1, 2, 4 is provided, respectively.(iii)That there does not exist Rota-Baxter operator of rank 3 with weight zero on the simple 3-Lie algebra is proved.(iv)Respectively,the Rote-Baxter operators of rank 1, 2, 4 with weight zero are proved and their concrete representations are given.