From $\mathcal{O}$-operators of 3-Dimensional 3-Lie algebras to 3-Pre-Lie algebras

Abstract

In this paper, we explicitly determine all $\mathcal{O}$-operators with respect to the adjoint representation of 3-dimensional complex 3-Lie algebras. Furthermore, we provide the induced 3-Pre-Lie algebra structures and the corresponding solutions of the 3-Lie classical Yang-Baxter equation in the 6-dimensional 3-Lie algebras $A{⋉}_{{\mathrm{ad}}^{\ast }}{A}^{\ast }$.