Central extensions of axial algebras

TL;DR

This paper extends the method of Skjelbred-Sund to construct and analyze central extensions of axial algebras, proving splitting results for complex simple Jordan cases and classifying low-dimensional examples.

Contribution

It introduces a new adaptation of the Skjelbred-Sund method for axial algebras and provides classification and splitting results for specific cases.

Findings

All axial central extensions of complex simple finite-dimensional Jordan algebras are split.

Non-split axial central extensions of dimension ≤ 4 are Jordan.

A classification of 2-dimensional axial algebras is provided.

Abstract

In this article, we develop a further adaptation of the method of Skjelbred-Sund to construct central extensions of axial algebras. We use our method to prove that all axial central extensions (with respect to a maximal set of axes) of complex simple finite-dimensional Jordan algebras are split and that all non-split axial central extensions of dimension $n\leq 4$ over an algebraically closed field of characteristic not $2$ are Jordan. Also, we give a classification of $2$-dimensional axial algebras and describe some important properties of these algebras.