Homogeneous Rota–Baxter operators on the $3$-Lie algebra $A_{\omega }$
Volume 148 / 2017
Abstract
We study homogeneous Rota–Baxter operators with weight zero on an infinite-dimensional simple $3$-Lie algebra $A_{\omega }$ over a field $ F$ ($\mathop {\rm ch}\nolimits F=0$), which is constructed from an associative commutative algebra $A$ with a derivation $\varDelta $ and an involution $\omega $ (Lemma 2.4). A homogeneous Rota–Baxter operator on $A_{\omega }$ is a linear map $R$ of $A_{\omega }$ satisfying $R(L_m)=f(m)L_m$ for all generators of $A_{\omega }$, where $f : \mathbb Z \rightarrow F$ is a function. We prove that $R$ is a homogeneous Rota–Baxter operator on $A_{\omega }$ if and only if $R$ is one of the five possibilities $R_{0_1}$, $R_{0_2}$, $R_{0_3}$, $R_{0_4}$ and $R_{0_5}$, described in Theorems 3.2, 3.12, 3.15, 3.19 and 3.21. Using the operators $R_{0_i}$, we construct new $3$-Lie algebras $(A, [ \,,\, ,\, ]_i)$ for $1\leq i\leq 5$, such that $R_{0_i}$ is a homogeneous Rota–Baxter operator on the $3$-Lie algebra $(A, [\, ,\, ,\, ]_i)$.
