symmetric monoidal (∞,1)-category of spectra
In this sense real normed division algebras may be thought of as a natural generalization of the more familiar real and complex numbers.
Moreover, if one regards the real numbers as a star-algebra, then each step in this sequence is given by applying the Cayley-Dickson construction. Applied to the octonions it yields the sedenions, which however are no longer a division algebra.
The Hopf invariant one theorem says that the only continuous functions between spheres of the form whose Hopf invariant is equal to 1 are the Hopf constructions on the four real normed division algebras, namely
Patterns related to Majorana spinors in spin geometry are intimately related to the four normed division algebras, and, induced by this, so is the classification of supersymmetry in the form of super Poincaré Lie algebras and super Minkowski spacetimes (which are built from these real spin representations). For more on this see at supersymmetry and division algebras.
(Moreover, apparently these two items are not unrelated, see here.)
A normed division algebra is
While the norm in a Banach algebra is in general only submultiplicative (), the norm in a normed division algebra must be multiplicative (). Accordingly, this norm is considered to be an absolute value and often written instead of . There is also a converse: if the norm on a Banach algebra is multiplicative (including ), then it must be a division algebra. While the term ‘normed division algebra’ does not seem to include the completeness condition of a Banach algebra, in fact the only examples have finite dimension and are therefore complete.
Accordingly, a normed division algebras is in particular a division composition algebra.
The Cayley–Dickson construction applies to an algebra with involution; by the abstract nonsense of that construction, we can see that the four normed division algebras above have these properties:
It is in fact true that all unital normed division algebras over are already finite dimensional, by (Urbanik-Wright 1960) (the authors give a reference on a non-unital infinite-dimensional normed division algebra). Hence the Hurwitz theorem together with Urbanik-Wright 1960 says that the above four exhaust all real normed division algebras.
For purely inseparable characteristic 2 field extensions one can apparently get infinite-dimensional examples; see this MathOverflow answer for reference.
, the trivial group
, the special orthogonal group acting via its canonical representaiton on the 3-dimensional space of imaginary octonions;
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
|the real numbers|
|the complex numbers|
|the quaternions||little string|
|the octonions||heterotic/type II string|
|normed division algebra||Riemannian -manifolds||Special Riemannian -manifolds|
|real numbers||Riemannian manifold||oriented Riemannian manifold|
|complex numbers||Kähler manifold||Calabi-Yau manifold|
|quaternions||quaternion-Kähler manifold||hyperkähler manifold|
The classification of real divsion composition algebras is originally due (Hurwitz theorem) to
The alternative classification as real alternative division algebras is due to
General discussion includes includes
Leonard Tornheim, Normed fields over the real and complex fields, Michigan Math. J. Volume 1, Issue 1 (1952), 61-68. (Euclid)
Silvio Aurora, On normed rings with monotone multiplication, Pacific J. Math. Volume 33, Number 1 (1970), 15-20 (JSTOR)
The result about removing the assumption of finite-dimensionality from unital normed division algebras appears in:
Exposition with emphasis on the octonions is in