In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation \`a la Rosenfeld.
a “magic” approach to octonionic
Rosenfeld spaces
12 December 2022
Alessio Marrania, Daniele Corradettib, David Chesterc,
Raymond Aschheimc, Klee Irwinc
a Instituto de Física Teorica, Departamento de Física,
Universidad de Murcia, Campus de Espinardo, E-30100, Spain
email: alessio.marrani@um.es
b Universidade do Algarve, Departamento de Matemática,
Campus de Gambelas, 8005-139 Faro, Portugal
email: a55944@ualg.pt
c Quantum Gravity Research,
Topanga Canyon Rd 101 S., California CA 90290, USA
email: DavidC@QuantumGravityResearch.org;
Raymond@QuantumGravityResearch.org;
Klee@QuantumGravityResearch.org
In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all
real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over
the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfor-
tunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems
he had been using do not actually hold true in the case of algebras that are not alternative nor
power-associative. A more rigorous approach to the definition of all the planes presented more
than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within
the theory of coset manifolds, which we exploit in this work, by making use of all real forms
of Magic Squares of order three and two over Hurwitz normed division algebras and their split
versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset manifolds which
seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar
analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric
cosets which do not have any interpretation à la Rosenfeld.
1
arXiv:2212.06426v1 [math.RA] 13 Dec 2022
Contents
1 Introduction
3
2 Real forms of Magic Squares of order 3 and 2
4
2.1 Hurwitz algebras and their split versions . . . . . . . . . . . . . . . . . . . . . . .
4
2.2 Trialities and derivations . . .