Twilled 3-Lie Algebras, Generalized Matched Pairs of 3-Lie Algebras and O-operators

In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an L∞-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 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 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.