Structure on the Simple Canonical Nambu Rota–Baxter 3-Lie Algebra $$A_{\partial }$$ A ∂

Abstract

In this paper, we study the structure of simple canonical Nambu 3-Lie algebra $$A_{\partial }=\sum \nolimits _{m\in Z} F z\exp (mx) \oplus \sum \nolimits _{m\in Z}F y\exp (mx)$$ . 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 0 satisfying $$R(L_m)=f(m+k)L_{m+k}$$ , $$R(M_m)=g(m+k)M_{m+k}$$ for all generators $$\{ L_m=z\exp (mx),$$ $$ M_m= y\exp (-mx)~~| ~~m\in Z\}$$ , where $$f, g : A_{\partial } \rightarrow F$$ are functions and $$k\in Z$$ . We obtain that R is a k-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ of weight 1 with $$k\ne 0$$ if and only if $$R=0$$ , and R is a 0-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ of weight 1 if and only if R is one of the ten possibilities described in Theorems 2.4 and 2.8; R is a k-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ of weight 0 with $$k\ne 0$$ if and only if R satisfies Theorem 3.1; and R is a 0-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ of weight 0 if and only if R is one of the four possibilities described in Theorem 3.3