In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_{\infty }$-algebra, whose Maurer–Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie 3-algebra whose Maurer–Cartan elements are $\mathcal{O}$-operators (also called relative Rota–Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of twilled 3-Lie algebras, which we call strict twilled 3-Lie algebras. Finally, we use $\mathcal{O}$-operators to construct explicit twilled 3-Lie algebras, and explain why an $r$-matrix for a 3-Lie algebra cannot give rise to a double construction 3-Lie bialgebra. Examples of twilled 3-Lie algebras are given to illustrate the various interesting phenomenon.

在本文中，首先我们介绍斜纹3-Lie代数的概念，并构造一个 ${\mathrm{大号}}_{\infty }$-代数，其Maurer-Cartan元素通过扭曲产生了新的斜纹3列代数。特别是，我们恢复了Maurer-Cartan元素为$\mathcal{Ø}$-3李代数上的-算子（也称为相对Rota-Baxter算子）。然后，我们使用3-Lie代数的广义表示形式引入3-Lie代数的广义匹配对的概念，这将产生斜纹3-Lie代数。通常匹配的3-Lie代数对对应于一类特殊的斜纹3-Lie代数，我们称其为严格的斜纹3-Lie代数。最后，我们使用$\mathcal{Ø}$-运算符构造显式斜纹3-Lie代数，并解释为什么 $\mathrm{[R}$-3-Lie代数的-矩阵不能产生双重构造的3-Lie双代数。给出了斜纹3-Lie代数的示例，以说明各种有趣的现象。