# Modules over axial algebras

## TL;DR

This paper introduces modules over axial algebras, providing new tools to analyze their structure, especially focusing on Matsuo algebras and their connection to 3-transposition groups, including a universal group construction.

## Contribution

It develops the theory of modules over axial algebras and links Matsuo algebras to 3-transposition groups, introducing a universal 3-transposition group from Fischer spaces.

## Key findings

- All known axial algebras fit into the module framework.
- Modules over Matsuo algebras relate to 3-transposition group representations.
- A universal 3-transposition group can be constructed from Fischer spaces.

## Abstract

We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group.