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

TL;DR

This paper introduces twilled 3-Lie algebras and generalized matched pairs, constructs related $L_$-algebras, and explores their connections with O-operators and $r$-matrices, expanding the understanding of 3-Lie algebra structures.

Contribution

It defines twilled 3-Lie algebras, constructs an $L_$-algebra framework, and links these to O-operators and generalized matched pairs, providing new insights and explicit examples.

Findings

Maurer-Cartan elements generate new twilled 3-Lie algebras

O-operators relate to Lie 3-algebras and twilled structures

Explicit constructions illustrate the theoretical concepts

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.