The Jordan pair content of the magic square and the geometry of the scalars in \(N=2\) supergravity
It is shown that three columns of the Freudenthal-Tits magic square (which is a table of 16 Lie algebras obtained by applying the well known Tits construction to the Jordan algebras of the Hermitian \(3\times 3\) matrices over \({\mathbb{R}}\), \({\mathbb{C}}\), \({\mathbb{H}}\), \({\mathbb{O}}\), the reals, complexes, quaternions, and octonions, respectively) can be characterized as algebras of the automorphism groups of, respectively, trace Jordan algebras (first row), Jordan pairs (second row), ternary algebras (third row). Of these results, only the one concerning the second row of the magic square is new, but putting them together suggests that an algebraic characterization of the fourth row is yet to be found. The approach is later on compared with that of Gürsey, and it is shown that the Hermitian symmetric spaces parametrized by the scalars of \(N=2\), \(d=4\) supergravity theories are related either to Jordan pairs or to geometries of projective dimension two, whose elements belong to a Jordan pair.
scientific article; zbMATH DE number 3922921
scientific article