Three Dixon-Rosenfeld Planes

TL;DR

This paper extends Rosenfeld's projective planes using the Dixon algebra, presenting three new coset manifolds with isometry algebras derived from Tits' magic formula, connecting them to exceptional Lie algebras.

Contribution

It introduces three Dixon-Rosenfeld planes with unique isometry algebras, expanding the framework of generalized projective planes beyond previous models.

Findings

Presented three new Dixon-Rosenfeld planes.

Derived their isometry algebras from Tits' magic formula.

Connected these planes to exceptional Lie algebras.

Abstract

Rosenfeld postulated ``generalized'' projective planes, which exploit a correspondence between rank-one idempotents of Jordan algebras $\mathfrak{J}_3(\mathbb{A})$ and points of projective planes $\mathbb{A}P^2$. The isometry groups of the generalized projective planes (which were later defined rigorously as homogeneous spaces) are entries of the Tits-Freudenthal magic square. Given recent interest in the Dixon algebra $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, we extend Rosenfeld's approach and present three new coset manifolds. These "Dixon-Rosenfeld planes" have isometry algebras that are obtained from Tits' magic formula and involve all tensorial components of the Dixon algebra. We show that these are the only three planes obtainable with Tits' formula that preserve the analogy with Rosenfeld's planes. These non-simple Lie algebras generalize $\mathfrak{f}_{4},\mathfrak{e}_{6},\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$ for the octonionic plane $\mathbb{O}P^{2}$ and in the octonionic Rosenfeld planes $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$, $\left(\mathbb{H}\otimes\mathbb{O}\right)P^{2}$ and $\left(\mathbb{O}\otimes\mathbb{O}\right)P^{2}$. We finally investigate the relationships between the isometry algebras of the Dixon-Rosenfeld planes and the exceptional Lie algebras.