Three Dixon-Rosenfeld Planes
David Chester, Alessio Marrani, Daniele Corradetti, Raymond Aschheim
TL;DR
This work extends Rosenfeld’s generalized projective planes to the non-Hurwitz Dixon algebra by applying the Tits-Freudenthal magic square to tensor components of $\mathbb{T}=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. It defines three inequivalent Lie algebras $\mathfrak{A}_1$, $\mathfrak{A}_2^{\text{enh}}$, $\mathfrak{A}_3$ and builds three 128-dimensional coset manifolds, the Dixon-Rosenfeld planes, as homogeneous spaces with these isometry algebras. The paper analyzes how these planes relate to the classical Rosenfeld planes and to exceptional Lie algebras, providing explicit embeddings and showing the planes are non-symmetric. It also connects two realizations of the Dixon algebra to potential physical applications, notably in the context of three generations and automorphism-like structures for $3\times3$ Hermitian matrices over $\mathbb{T}$.
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.