Central extensions of axial algebras

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 $n\le 4$ 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.