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Abstract

We study $k$-order homogeneous Rota–Baxter operators of weight $1$ on the simple $3$-Lie algebra $A_{\omega }$ (over a field $\mathbb F$ of characteristic zero), which is realized by an associative commutative algebra $A$ equipped with a derivation $\varDelta$ and an involution $\omega$ (Lemma 2.3). A $k$-order homogeneous Rota–Baxter operator on $A_{\omega }$, where $k\in \mathbb Z$, is a Rota–Baxter operator $R$ satisfying $R(L_m)=f(m+k)L_{m+k}$ for all generators $\{ L_m \mid m\in \mathbb Z \}$ of $A_{\omega }$ and a map $f : \mathbb Z \rightarrow \mathbb F$. We prove that $R$ is a $k$-order homogeneous Rota–Baxter operator on $A_{\omega }$ of weight $1$ with $k\not =0$ if and only if $R=0$ (Theorem 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 thirty-eight possibilities which are described in Theorems 3.5, 3.7, 3.9, 3.10, 3.18, 3.21 and 3.22.