Homogeneous Rota-Baxter Operators On 3-Lie Algebra

In this paper, the structure of homogeneous Rota-Baxter operator of weight zero and weight one on the infinite dimensional 3-Lie algebra over a field F (chF=0) are studied. A homogeneous Rota-Baxter operator on the 3-Lie algebra Aw is a Rota-Baxter opera-tor Rk, k?Z satisfying that there exists f:Z?F, satisfies that Rk(Lm)=f(m+k)Lm-k. The structure of homogeneous Rota-Baxter operator of weight zero is completely studied. The sufficient and necessary conditions for a linear map being a homogeneous Rota-Baxter oper-ator of weight zero are provided, and concrete expressions are given. The new Rota-Baxter 3-Lie algebras are constructed by all homogeneous Rota-Baxter operators. The structure of homogeneous Rota-Baxter operator of weight one is completely studied. It is proved that the homogeneous of Rota-Baxter operator of weight one with k?0 is only zero operator. And the sufficient and necessary condition for a linear map being a homogeneous Rota-Baxter operator of weight one with k=0 are provided.