symmetric monoidal (∞,1)-category of spectra
While the norm in a Banach algebra is normally 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.
The only normed division algebra over the complex numbers is the algebra of complex numbers themselves. Up to isomorphism, there are exactly four finite-dimensional normed division algebras over the real numbers:
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:
A normed field is a commutative normed division algebra; it follows from the preceding that the only normed fields are and .
|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|