Space–times over normed division algebras, revisited

Riemannian Geometry over different Normed Division Algebras

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkähler and G2-man...

Triangulaizability of Algebras Over Division Rings

triangulaizability of algebras over division rings

Normed division algebras with R: introducing the onion package

where multiplication is denoted by juxtaposition; note the absence of associativity. A division algebra is a nontrivial algebra in which division is possible: for any a ∈ V and any nonzero b ∈ V , there exists precisely one element x with a = bx and precisely one element y with a = yb. A normed division algebra is a division algebra with a norm ||·|| satisfying ||xy|| = ||x|| ||y||. There are p...

Quadratic Division Algebras Revisited ( Remarks On

In his remarkable article “Quadratic division algebras” (Trans. Amer. Math. Soc. 105 (1962), 202–221), J. M. Osborn claims to solve ‘the problem of determining all quadratic division algebras of order 4 over an arbitrary field F of characteristic not two . . . modulo the theory of quadratic forms over F ’ (cf. p. 206). While we shall explain in which respect he has not achieved this goal, we sh...