# Generalized representations of 3-Hom-Lie algebras

## TL;DR

This paper extends the concept of generalized representations from 3-Lie algebras to 3-Hom-Lie algebras, developing cohomology theory, semi-direct products, and exploring extensions.

## Contribution

It introduces generalized representations for 3-Hom-Lie algebras, constructs their cohomology, and links extensions to semidirect products.

## Key findings

- Developed cohomology theory for 3-Hom-Lie algebras
- Computed 2-cocycles in the new cohomology
- Established connections between extensions and semidirect products

## Abstract

The propose of this paper is to extend generalized representations of 3-Lie algebras to Hom-type algebras. We introduce the concept of generalized representation of multiplicative 3-Hom-Lie algebras, develop the corresponding cohomology theory and study semi-direct products. We provide a key construction, various examples and computation of 2-cocycles of the new cohomology. Also, we give a connection between a split abelian extension of a 3-Hom-Lie algebra and a generalized semidirect product 3-Hom-Lie algebra.