Homogenous Rota-Baxter 3-Lie Algebras

In this paper,we mainly study the structures of homogenous Rota-Baxter 3-Lie algebras which are constructed by the infinite dimensional simple 3-Lie algebra Aω and its homogenous Rota-Baxter operators of weight one,where Aω is an infinite dimensional 3-Lie algebra on the vector space(?)with a basis {Lm|m∈Z},F is a field of characteristic zero.The homogenous Rota-Baxter operator R of weight λ on the 3-Lie algebra Aω is a Rota-Baxter operator of weight κ satisfying identity R(Lm)= f(m +k)Lm+k,where λ∈ F,k ∈ Z and f:Z→ F is a function.Since the Rota-Baxter operators of weight A(λ≠0)on a 3-Lie algebra is completely determined by the case A = 1,we study k-order homogenous Rota-Baxter operators of weight one on the 3-Lie algebra Aω by two cases 1)f(0)+ f(1)+1 ≠ 0,2)f(0)+ f(1)+ 1= 0 and f(0)≠ 0.It is proved that there exists only zero operator when k ≠0.And in the case k =0,we provide the concrete expression of 20 homogenous Rota-Baxter operators and the affirmative value of fi,1 ≤i≤20 which satisfy Ri(Lm)= fi(m)Lm.On the base vector space A of the 3-Lie algebra Aω,we construct eighteen classes of 3-Lie algebras(A,[,,]j),1 ≤ j ≤ 20,j ≠ 2,18,and prove that Rj is the homogenous Rota-Baxter operator on(A,[,,]j),respectively,and therefore,(A,[,,]1 ≤ j<20,j ≠ 2,18 are homogenous Rota-Baxter 3-Lie algebras.