Axial algebras of Jordan and Monster type

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.

On primitive 3-generated axial algebras of Jordan type

“…Axial algebras were introduced by Hall, Rehren, and Shpectorov in [6,7]. Current state-of-art of this topic can be found in a recent survey [10].…”

“…This implies that the class of primitive PC(0)-axial algebras is exactly the intersection of axial algebra classes corresponding to two fusion laws in Table 1: PC(0) and J ( 12 ). Note that until now, most of the papers on axial algebras have been devoted to two cases: axial algebras of Jordan type (fusion laws J (η)) and Monster type [10]. In the proof of Theorem 1.1, we use methods developed earlier for axial algebras of Jordan type and show how they can be applied to other classes of algebras.…”