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.
This classification turns out to closely connect to various other systems of exceptional structures in mathematics and physics. Notably 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 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.
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:
The automorphism groups of the real normed division algebras are
, 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|
The classification of real divsion composition algebras is originally due (Hurwitz theorem) 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)
Exposition with emphasis on the octonions is in