Return to Revisions

The appropriate Jordan algebra is $J_n(\mathbb{H}) := \{ A \in M(n,\mathbb{H})\\,|\\, \overline{A}^T=A \}$ with multiplication defined as $A\circ B = (AB+BA)/{2}$. The metric is defined by $\mathrm{Tr}(A^2)$. The quaternionic projective space can be then defined as the space of elements of $J_n(\mathbb{H})$ of rank one and trace one (think of these matrices as projectors to one-dimensional subspaces) or alternatively as the (real) projectivization of rank one matrices.