Variety of Jordan algebras in small dimensions

Abstract

The variety \(\,{\mathcal{J}} or_n\,\) of Jordan unitary algebra structures on \(\,{\bf k}^n\,\), \(\,{\bf k}\,\) an algebraically closed field with char\(\,{\bf k}\neq 2\,\), is studied, as well as infinitesimal deformations of Jordan algebras. Also we establish the list of \(\,{\rm GL}_n\)-orbits on \(\,{\mathcal{J}} or_n\), \(\,n=4,5\,\) under the action of structural transport. The numbers \(\,jor_4\,\) and \(\,jor_5\,\) of irreducible components are \(\,3\,\) and \(\,6\,\) respectively; a list of generic structures is included.