Homogeneous Rota-Baxter operators on $A_{\omega}$ (II)

In this paper we study $k$-order homogeneous Rota-Baxter operators with weight $1$ on the simple $3$-Lie algebra $A_{\omega}$ (over a field of characteristic zero), which is realized by an associative commutative algebra $A$ and a derivation $\Delta$ and an involution $\omega$ (Lemma \mref{lem:rbd3}). A $k$-order homogeneous Rota-Baxter operator on $A_{\omega}$ is a linear map $R$ satisfying $R(L_m)=f(m+k)L_{m+k}$ for all generators $\{ L_m~ |~ m\in \mathbb Z \}$ of $A_{\omega}$ and a map $f : \mathbb Z \rightarrow\mathbb F$, where $k\in \mathbb Z$. We prove that $R$ is a $k$-order homogeneous Rota-Baxter operator on $A_{\omega}$ of weight $1$ with $k\neq 0$ if and only if $R=0$ (see Theorems 3.2, and $R$ is a $0$-order homogeneous Rota-Baxter operator on $A_{\omega}$ of weight $1$ if and only if $R$ is one of the forty possibilities which are described in Theorems3.5, 3.7, 3.9, 3.10, 3.18, 3.21 and 3.22.