symmetric monoidal (∞,1)-category of spectra
A normed division algebra is a not-necessarily associative algebra, over some ground field, that is both a division algebra as well as a multiplicatively normed algebra.
It turns out that over the real numbers there are precisely only four normed division algebras up to isomorphism: the algebras of
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.
This classification turns out to closely connect to various other systems of exceptional structures in mathematics and physics:
The Hopf invariant one theorem says that the only continuous functions between spheres of the form $S^{2n-1}\to S^n$ 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
that is also a Banach algebra.
While the norm in a Banach algebra is in general only submultiplicative (${\|x y\|} \leq {\|x\|} {\|y\|}$), the norm in a normed division algebra must be multiplicative (${\|x y\|} = {\|x\|} {\|y\|}$). 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 ${\|1\|} = 1$), 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.
Over the complex numbers, the only normed division algebra is the algebra of complex numbers themselves.
The Hurwitz theorem says that over the real numbers there are, up to isomorphism, exactly four finite-dimensonal normed division algebras :
In fact these are also exactly the real alternative division algebras:
The only division algebras over the real numbers which are also alternative algebras are the real numbers themselves, the complex numbers, the quaternions and the octonions.
(Zorn 30).
Each of these is produced from the previous one by the Cayley–Dickson construction; when applied to $\mathbb{O}$, this construction produces the algebra of sedenions, which do not form a division 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:
However, these algebras do all have some useful algebraic properties; in particular, they are all alternative (a weak version of associativity). They are also all composition algebras.
A normed field is a commutative normed division algebra; it follows from the preceding that the only normed fields over $\mathbb{R}$ are $\mathbb{R}$ and $\mathbb{C}$ (e.g. Tornheim 52).
It is in fact true that all unital normed division algebras over $\mathbb{R}$ 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 automorphism groups of the real normed division algebras, as normed algebras, are
$Aut(\mathbb{R}) = 1$, the trivial group
$Aut(\mathbb{C}) = \mathbb{Z}/2$ the group of order 2, acting by complex conjugation;
$Aut(\mathbb{H}) = SO(3)$, the special orthogonal group acting via its canonical representaiton on the 3-dimensional space of imaginary octonions;
$Aut(\mathbb{O}) = G_2$, the exceptional Lie group G2.
The Hopf invariant one theorem says that the spheres carrying H-space structure are precisely the unit spheres in one of the normed division algebras
(Adams 60)
The Freudenthal magic square is a special square array of Lie algebras/Lie groups labeled by pairs of real normed division algebras and including all the exceptional Lie groups except G2.
exceptional spinors and real normed division algebras
Lorentzian spacetime dimension | $\phantom{AA}$spin group | normed division algebra | $\,\,$ brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\phantom{A}$ $\mathbb{R}$ the real numbers | super 1-brane in 3d |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\phantom{A}$ $\mathbb{C}$ the complex numbers | super 2-brane in 4d |
$6 = 5+1$ | $Spin(5,1) \simeq SL(2, \mathbb{H})$ | $\phantom{A}$ $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}$ | $\phantom{A}$ $\mathbb{O}$ the octonions | heterotic/type II string |
see division algebra and supersymmetry
normed division algebra | $\mathbb{A}$ | Riemannian $\mathbb{A}$-manifolds | Special Riemannian $\mathbb{A}$-manifolds |
---|---|---|---|
real numbers | $\mathbb{R}$ | Riemannian manifold | oriented Riemannian manifold |
complex numbers | $\mathbb{C}$ | Kähler manifold | Calabi-Yau manifold |
quaternions | $\mathbb{H}$ | quaternion-Kähler manifold | hyperkähler manifold |
octonions | $\mathbb{O}$ | Spin(7)-manifold | G2-manifold |
(Leung 02)
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
Discussion of Riemannian geometry and special holonomy modeled on the different normed division algebras is in