Representations and cohomologies of differential 3-Lie algebras with any weight

TL;DR

This paper develops the theory of representations and cohomologies for differential 3-Lie algebras with arbitrary weight, exploring their structures, relationships, and extensions.

Contribution

It introduces the concept of representations and cohomology for differential 3-Lie algebras and examines their deformations, O-operators, and abelian extensions.

Findings

Established a cohomology theory for differential 3-Lie algebras

Linked cohomologies of differential 3-Lie algebras with their associated Leibniz algebras

Analyzed deformations and abelian extensions of these algebras

Abstract

The purpose of the present paper is to study representations and cohomologies of differential 3-Lie algebras with any weight. We introduce the representation of a differential 3-Lie algebra. Moreover,we develop cohomology theory of a differential 3-Lie algebra. We also depict the relationship between the cohomologies of a differential 3-Lie algebra and its associated differential Leibniz algebra with weight zero. The deformation and O-operator of differential 3-Lie algebras are also investigated. Finally, we consider abelian extensions of differential 3-Lie algebras.