3-Lie Algebra A_p-modules And Homogeneous Rota-Baxter Operators

Rota-Baxter operators on 2-ary algebras have wide applications in many fields of math-ematics and physics.In recent years,Rota-Baxter operators on n-ary algebras are studied by many mathematicians.Inspired by that,we study the structure of Rota-Baxter operators of simple canonical Nambu 3-Lie algebra Ap =∑m∈Z Fz exp(mx)(?)∑m∈Z Fy exp(mx).We pay close attention to a special class of Rota-Baxter operators,which are k-order homogeneous Rota-Baxter operators R of weight 1 and weight zero which satisfy R(Lm)= f(m+k)Lm+k,R(Mm)=g(m+k)Mm+k for all generators{Lm=z exp(mx),Mm=yexp(-mx)|m ∈Z｝,where f,g:Ap→F are functions and k E Z.We obtain that R is a k-order homogeneous Rota-Baxter operator on Ap of weight 1 with k ≠4 0 if and only if R,0,and R is a 0-order homogeneous Rota-Baxter operator on Ap of weight 1 if and only if R is one of the ten possi-bilities described in Theorems 3.4 and 3.8;R is a k-order homogeneous Rota-Baxter operator on A,of weight 0 with k≠0 if and only if R satisfies Theorem 4.1;and R is a 0-order homo-geneous Rota-Baxter operator on Ap of weight 0 if and only if R is one of the four possibilities described in Theorem 4.3.At last of the paper,the 3-Lie algebra Ap-module Tμυ is constructed,where μ∈ F,υ= 0 or 1.And the relations between the 3-Lie algebra Ap-modules(Y,ρ)and the induced modules(V,ρ)of the inner derivation algebra ad(Aρ)are discussed.