Universal decomposition algebras and the classification of 2-generated non-primitive axial algebras of Jordan type

TL;DR

This paper introduces universal decomposition algebras and uses them to classify 2-generated non-primitive axial algebras of Jordan type, expanding the understanding of these algebraic structures.

Contribution

It constructs universal algebras for these classes and provides a classification of specific 2-generated non-primitive axial algebras of Jordan type.

Findings

Existence of universal algebras for decomposition and axial decomposition classes

Classification of 2-generated non-primitive axial algebras of Jordan type

Enhanced understanding of algebraic structures related to axial algebras

Abstract

Decomposition algebras and axial decomposition algebras are classes of commutative nonassociative algebras which are generalizations of axial algebras. The classes decomposition algebras, axial decomposition al;gebras and non-primitive axial algebras also have univesal algebras. Furthermore, by using the existence of a universal algebra, 2-generated non-primitive axial algebras of Jordan type is classified.