Homogeneous Rota-Baxter operators on 3-Lie algebra Aω

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In the paper we study homogeneous Rota-Baxter operators with weight zero on the infinite dimensional simple 3-Lie algebra Aω over a field F (ch F=0) which is realized by an associative commutative algebra A and a derivation Δ and an involution ω (Lemma lem:rbd3). A homogeneous Rota-Baxter operator on Aω is a linear map R of Aω satisfying R(Lₘ)=f(m)Lₘ for all generators of Aω, where f: Aω → F. We proved that R is a homogeneous Rota-Baxter operator on Aω if and only if R is the one of the five possibilities R₀₁, R₀₂,R₀₃,R₀₄ and R₀₅, which are described in Theorem thm:thm1, thm:thm4, thm:thm01, thm:thm03 and thm:thm04. By the five homogeneous Rota-Baxter operators R₀ᵢ, we construct new 3-Lie algebras (A, [,,]ᵢ) for 1≤ i≤ 5, such that R₀ᵢ is the homogeneous Rota-Baxter operator on 3-Lie algebra (A, [,,]ᵢ), respectively.