Cohomologies of 3-Lie algebras with derivations

TL;DR

This paper develops a cohomology theory for 3-Lie algebras with derivations, explores their deformations and extensions, and introduces categorified structures, establishing classification results for skeletal 3-Lie2Der pairs.

Contribution

It introduces a cohomology framework for 3-LieDer pairs, defines 3-Lie2Der pairs, and classifies skeletal 3-Lie2Der pairs via triples involving cocycles and crossed modules.

Findings

Cohomology for 3-LieDer pairs is established.

Classification of skeletal 3-Lie2Der pairs is achieved.

Correspondence between strict 3-Lie2Der pairs and crossed modules is proven.

Abstract

In this paper, we consider a 3-Lie algebra with a derivation (called a 3-LieDer pair). We define cohomology for a 3-LieDer pair with coefficients in a representation. We use this cohomology to study deformations and abelian extensions of 3-LieDer pairs. We give the notion of a 3-Lie2Der pair, which can be viewed as the categorification of a 3-LieDer pair. We show that skeletal 3-Lie2Der pairs are classified by triples given by 3-LieDer pairs, representations and 3-cocycles. We define crossed modules of 3-LieDer pairs and show that there exists a one-to-one correspondence between strict 3-Lie2Der pairs and crossed modules of 3-LieDer pairs.