Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators

Abstract

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 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 can not give rise to a double construction 3-Lie bialgebra. Examples of twilled 3-Lie algebras are given to illustrate the various interesting phenomenon.

19 pages, comments are welcome

References in corpus (2)

• n-ary algebras: a review with applications
• Derived bracket construction up to homotopy and Schroder numbers

Cited by in corpus (1)

• $3$-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras