$3$-$Lie_\infty$-algebras and $3$-Lie 2-algebras

TL;DR

This paper introduces and studies higher homotopy and categorified versions of 3-Lie algebras, establishing their equivalence and exploring specific subclasses like skeletal and strict 3-Lie 2-algebras.

Contribution

It defines 3-Lie_infty-algebras and 3-Lie 2-algebras, proves their categorical equivalence, and constructs examples from symplectic 3-Lie algebras.

Findings

Equivalence between 2-term 3-Lie_infty-algebras and 3-Lie 2-algebras

Detailed study of skeletal and strict 3-Lie 2-algebras

Construction of 3-Lie 2-algebras from symplectic 3-Lie algebras

Abstract

In this paper, we introduce the notions of a $3$-$Lie_\infty$-algebra and a 3-Lie 2-algebra. The former is a model for a 3-Lie algebra that satisfy the fundamental identity up to all higher homotopies, and the latter is the categorification of a 3-Lie algebra. We prove that the 2-category of 2-term $3$-$Lie_\infty$-algebras is equivalent to the 2-category of 3-Lie 2-algebras. Skeletal and strict 3-Lie 2-algebras are studied in detail. A construction of a 3-Lie 2-algebra from a symplectic 3-Lie algebra is given.