Axial algebras of Jordan and Monster type

TL;DR

This survey explores axial algebras, focusing on Jordan and Monster types, highlighting their structural properties and connections to simple groups and automorphism groups.

Contribution

It introduces axial algebras, details their properties, and emphasizes the significance of Jordan and Monster types in relation to simple groups.

Findings

Axial algebras are linked to group theory through their automorphisms.

Jordan and Monster type algebras are rich in examples related to simple groups.

The survey clarifies the structural properties of these algebras.

Abstract

Axial algebras are a class of non-associative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is graded, the algebra has a naturally associated group of automorphisms and thus axial algebras are inherently related to group theory. Examples include most Jordan algebras and the Griess algebra for the Monster sporadic simple group. In this survey, we introduce axial algebras, discuss their structural properties and then concentrate on two specific classes: algebras of Jordan and Monster type, which are rich in examples related to simple groups.