Homogeneous Rota–Baxter operators on the $3$-Lie algebra $A_{\omega }$

Ruipu Bai¹ ; Yinghua Zhang¹

¹ College of Mathematics and Information Science Hebei University Baoding 071002, China

Colloquium Mathematicum, Tome 148 (2017) no. 2, pp. 195-213.

Résumé

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 $(A, [\, ,\, ,\, ]_i)$.

Détail

DOI : 10.4064/cm6829-2-2016

Keywords: study homogeneous rota baxter operators weight zero infinite dimensional simple lie algebra omega field mathop nolimits which constructed associative commutative algebra derivation vardelta involution nbsp omega lemma nbsp homogeneous rota baxter operator omega linear map omega satisfying m generators omega where mathbb rightarrow function prove homogeneous rota baxter operator omega only five possibilities described theorems using operators construct lie algebras leq leq homogeneous rota baxter operator lie algebra