Axes of Jordan type in non-commutative algebras

TL;DR

This paper explores non-commutative decomposition algebras that extend the concept of primitive axial algebras of Jordan type, linking them to broader algebraic structures like 3-transposition groups and vertex operator algebras.

Contribution

It introduces and studies non-commutative analogues of primitive axial algebras of Jordan type, expanding the framework of axial algebras to include non-commutative decomposition algebras.

Findings

Established connections between non-commutative decomposition algebras and 3-transposition groups.

Extended the theory of axial algebras to non-commutative settings.

Provided structural insights into non-commutative generalizations of Jordan-type algebras.

Abstract

The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, withtheir introduction of {\it axial algebras}, and in particular {\it primitive axial algebras of Jordan type} (PJs for short). It turns out that these notions are closely related to $3$-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov, and M. Van Couwenberghe generalized axial algebrasto {\it decomposition algebras} which, in particular, are not necessarily commutative. This paper deals with decomposition algebras which are non-commutative versions of PJs.