A new approach to representations of $3$-Lie algebras and abelian extensions

TL;DR

This paper introduces generalized representations and cohomology for 3-Lie algebras, establishing a framework for abelian extensions and their classification via Maurer-Cartan elements.

Contribution

It develops a new notion of generalized representation for 3-Lie algebras, constructs a corresponding cohomology theory, and characterizes abelian extensions using Maurer-Cartan elements.

Findings

Defined generalized representations of 3-Lie algebras

Constructed a cohomology theory for these representations

Characterized abelian extensions via Maurer-Cartan elements

Abstract

In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.