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arXiv · 2010.04881

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

Shuai Hou·Yunhe Sheng·Rong Tang
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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.

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Shuai Hou, Yunhe Sheng, Rong Tang. 2020-10-10. Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators. https://doi.org/10.1016/j.geomphys.2021.104148

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