Variety of Jordan algebras in small dimensions

Iryna Kashuba

Algebra and discrete mathematics, no. 2 (2006), pp. 62-76.

Résumé

The variety $\mathcal{J}o{r}_n$ of Jordan unitary algebra structures on ${\mathbf{k}}^{\mathbf{n}}$, $\mathbf{k}$ an algebraically closed field with char $\mathbf{k}\ne \mathbf{2}$, is studied, as well as infinitesimal deformations of Jordan algebras. Also we establish the list of ${\mathrm{G}\mathrm{L}}_{\mathrm{n}}$-orbits on $\mathcal{J}o{r}_n$, $n=4,5$ under the action of structural transport. The numbers $jo{r}_4$ and $jo{r}_5$ of irreducible components are 3 and 6 respectively; a list of generic structures is included.