Abstract:

Constructing 3-Pre-Lie algebras has always been a difficult problem; until now, there have been very few examples of 3-Pre-Lie algebras. In this paper, we use homogenous Rota-Baxter operators of weight zero on the infinite dimensional 3-Lie algebra $A_{\omega}=\langle L_m | m\in {\mathbb{Z}}\rangle$ to construct 3-Pre-Lie algebras $B_k,~0\leqslant k\leqslant 4$ , and we subsequently discuss the structure. It is shown that $B_2$ and $B_4$ are non-isomorphic simple 3-Pre-Lie algebras, $B_1$ is an indecomposable 3-Pre-Lie algebra with infinitely many one-dimensional ideals, and $B_3$ is an indecomposable 3-Pre-Lie algebra with finitely many ideals.

Key words: 3-Pre-Lie algebras, 3-Lie algebras, homogenous Rota-Baxter operator