# 3-dimensional algebras. Part 1. Skew-symmetric case

## TL;DR

This paper classifies all 3-dimensional skew-symmetric algebras over fields with characteristic not 2 and explores related Hom-Lie algebra structures, including some results on 4-dimensional cases.

## Contribution

It provides a complete classification of 3-dimensional skew-symmetric algebras and characterizes the subvariety of 3-dimensional Hom-Lie algebras, extending to 4-dimensional cases.

## Key findings

- Complete classification of 3D skew-symmetric algebras
- Identification of the subvariety of 3D Hom-Lie algebras
- Initial results on 4-dimensional cases

## Abstract

An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application. We determine all these algebras in dimension $3$ over a field of characteristic different from $2$. As an application, we determine the subvariety of $3$-dimensional Hom-Lie algebras. For this type of algebras, we study also the dimension $4$.