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\section*{normed division algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{normed_division_algebras}{}\section*{{Normed division algebras}}\label{normed_division_algebras}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Classification}{Classification}\dotfill \pageref*{Classification} \linebreak
\noindent\hyperlink{Automorphisms}{Automorphisms}\dotfill \pageref*{Automorphisms} \linebreak
\noindent\hyperlink{relation_to_hspace_structures_on_sphere_hopf_invariant_one}{Relation to H-space structures on sphere (Hopf invariant one)}\dotfill \pageref*{relation_to_hspace_structures_on_sphere_hopf_invariant_one} \linebreak
\noindent\hyperlink{magic_square}{Magic square}\dotfill \pageref*{magic_square} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A normed division algebra is a [[not-necessarily associative algebra]], over some [[ground field]], that is

\begin{enumerate}%
\item a [[division algebra]] $\big( \text{i.e.}\, (a \cdot b = 0) \Rightarrow (a = 0 \,\text{or}\, b = 0) \big)$


\item a multiplicatively [[normed algebra]] $\big( \text{i.e.}\, {\Vert a \cdot b\Vert}  \leq C \cdot  {\Vert a\Vert} \cdot {\Vert b\Vert} \big)$.



\end{enumerate}
It should be the case (at least maybe for finite-dimensional algebras) that the division property (1) implies that the norm property (2) holds in the stronger form

\begin{displaymath}
{\vert a \cdot b\vert} 
  \;=\;
  {\vert a \vert} \cdot {\vert b \vert}
\end{displaymath}
and this is how most (or all) authors actually define normed division algebras, and that's what we assume to be meant now.

$\,$

It turns out ([[Hurwitz' theorem]]) that over the [[real numbers]] there are precisely only four normed division algebras up to [[isomorphism]]: the algebras of

\begin{enumerate}%
\item [[real numbers]],


\item [[complex numbers]],


\item [[quaternions]],


\item [[octonions]].



\end{enumerate}
In this sense real normed division algebras may be thought of as a natural generalization of the [[real numbers]] and the [[complex numbers]].

Moreover, if one regards the [[real numbers]] as a [[star-algebra]] with trivial [[anti-involution]], then each step in the above sequence is given by applying the [[Cayley-Dickson construction]]. (While the process of applying the [[Cayley-Dickson construction]] continues, next with the[[sedenions]], these and the following are no longer [[division algebras]].)

This classification of real normed division algebras is closely related to various other systems of [[universal exceptionalism|exceptional]] structures in [[mathematics]] and [[physics]]:

\begin{itemize}%
\item The \emph{[[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

\begin{enumerate}%
\item the [[real Hopf fibration]];


\item the [[complex Hopf fibration]];


\item the [[quaternionic Hopf fibration]];


\item the [[octonionic Hopf fibration]].



\end{enumerate}

\item 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 \emph{[[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 \emph{[[supersymmetry and division algebras]]}.



\end{itemize}
(Moreover, apparently these two items are not unrelated, see [[schreiber:Equivariant cohomology of M2/M5-branes|here]].)

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{normed division algebra} is

\begin{itemize}%
\item a [[division algebra]];


\item that is also a [[Banach algebra]].



\end{itemize}
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 [[complete space|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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Classification}{}\subsubsection*{{Classification}}\label{Classification}

Over the [[complex numbers]], [[generalized the|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 :

\begin{itemize}%
\item $\mathbb{R}$, the algebra of [[real numbers]],
\item $\mathbb{C}$, the algebra of [[complex numbers]],
\item $\mathbb{H}$, the algebra of [[quaternions]],
\item $\mathbb{O}$, the algebra of [[octonions]].

\end{itemize}
In fact these are also exactly the real [[alternative algebra|alternative]] division algebras:

\begin{prop}
\label{ZornTheorem}\hypertarget{ZornTheorem}{}
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]].

\end{prop}
(\hyperlink{Zorn30}{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 \emph{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:

\begin{itemize}%
\item $\mathbb{R}$ is [[associative algebra|associative]], [[commutative algebra|commutative]], and with trivial involution,
\item $\mathbb{C}$ is associative and commutative but has nontrivial involution,
\item $\mathbb{H}$ is associative but noncommutative and with nontrivial involution,
\item $\mathbb{O}$ is neither associative, commutative, nor with trivial involution.

\end{itemize}
However, these algebras do all have some useful algebraic properties; in particular, they are all [[alternative algebra|alternative]] (a weak version of associativity). They are also all [[composition algebra]]s.

A \textbf{[[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. \hyperlink{Tornheim52}{Tornheim 52}).

It is in fact true that all \emph{unital} normed division algebras over $\mathbb{R}$ are already finite dimensional, by (\hyperlink{UrbanikWright60}{Urbanik-Wright 1960}) (the authors give a reference on a non-unital infinite-dimensional normed division algebra). Hence the [[Hurwitz theorem]] together with \hyperlink{UrbanikWright60}{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 \href{http://mathoverflow.net/a/45655/4177}{this MathOverflow answer} for reference.

\hypertarget{Automorphisms}{}\subsubsection*{{Automorphisms}}\label{Automorphisms}

The [[automorphism groups]] of the real normed division algebras, as [[normed algebras]], are

\begin{itemize}%
\item $Aut(\mathbb{R}) = 1$, the [[trivial group]]


\item $Aut(\mathbb{C}) = \mathbb{Z}/2$ the [[group of order 2]], acting by [[complex conjugation]];


\item $Aut(\mathbb{H}) = SO(3)$, the [[special orthogonal group]] acting via its canonical representation on the 3-dimensional space of imaginary quaternions;


\item $Aut(\mathbb{O}) = G_2$, the [[exceptional Lie group]] [[G2]].



\end{itemize}
Incidentally, there is a sense in which this sequence of groups continues, with the [[infinity-group]] [[G3]] (the [[Dwyer-Wilkerson H-space]]):

\begin{tabular}{l|l|l|l|l|l}
$n=$&0&1&2&3&4\\
\hline 
$DI(n)=$&[[trivial group&1]]&[[Z/2]]&[[SO(3)]]&[[G2]]\\
&= Aut(R)&\href{complex+number#AutomorphismsOfComplexNumbersIsZ2}{= Aut(C)}&\href{quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3}{= Aut(H)}&\href{octonion#AutomorphismsOfOctonionAlgebraIsG2}{= Aut(O)}&\\
\end{tabular}

\hypertarget{relation_to_hspace_structures_on_sphere_hopf_invariant_one}{}\subsubsection*{{Relation to H-space structures on sphere (Hopf invariant one)}}\label{relation_to_hspace_structures_on_sphere_hopf_invariant_one}

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



(\href{Hopf+invariant+one#Adams60}{Adams 60})

\hypertarget{magic_square}{}\subsubsection*{{Magic square}}\label{magic_square}

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]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hopf invariant one]]


\item [[Cayley–Dickson construction]]



\end{itemize}
[[!include exceptional spinors and division algebras -- table]]

see [[division algebra and supersymmetry]]

[[!include normed division algebra Riemannian geometry -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The classification of real divsion composition algebras is originally due ([[Hurwitz theorem]]) to

\begin{itemize}%
\item [[Adolf Hurwitz]], \emph{\"U{}ber die Composition der quadratischen Formen von beliebig vielen Variabeln}, Nachr. Ges. Wiss. G\"o{}ttingen (1898) 309--316

\end{itemize}
The alternative classification as real [[alternative algebra|alternative]] division algebras is due to

\begin{itemize}%
\item [[Max Zorn]], \emph{Theorie der alternativen Ringe}, Abhandlungen des Mathematischen Seminars der Universit\"a{}t Hamburg 8 (1930), 123-147

\end{itemize}
General discussion includes includes

\begin{itemize}%
\item Leonard Tornheim, \emph{Normed fields over the real and complex fields}, Michigan Math. J. Volume 1, Issue 1 (1952), 61-68. (\href{http://projecteuclid.org/euclid.mmj/1028989727}{Euclid})


\item Silvio Aurora, \emph{On normed rings with monotone multiplication}, Pacific J. Math. Volume 33, Number 1 (1970), 15-20 (\href{http://projecteuclid.org/euclid.pjm/1102977236}{JSTOR})



\end{itemize}
The result about removing the assumption of finite-dimensionality from unital normed division algebras appears in:

\begin{itemize}%
\item Kazimierz Urbanik and Fred B. Wright, \_ Absolute-valued algebras\_, Proc. Amer. Math. Soc. \textbf{11} (1960), 861-866, doi:\href{https://doi.org/10.1090/S0002-9939-1960-0120264-6}{10.1090/S0002-9939-1960-0120264-6}

\end{itemize}
Exposition with emphasis on the [[octonions]] is in

\begin{itemize}%
\item [[John Baez]], \emph{\href{http://math.ucr.edu/home/baez/octonions/node2.html}{Normed Division Algebras}}


\item [[John Baez]], \href{http://math.ucr.edu/home/baez/week59.html}{This Week's Finds --- Week 59}



\end{itemize}
Discussion of [[Riemannian geometry]] and [[special holonomy]] modeled on the different normed division algebras is in

\begin{itemize}%
\item [[Naichung Conan Leung]], \emph{Riemannian Geometry Over Different Normed Division Algebras}, J. Differential Geom. Volume 61, Number 2 (2002), 289-333. (\href{http://projecteuclid.org/euclid.jdg/1090351387}{euclid})

\end{itemize}
[[!redirects normed division algebra]] [[!redirects normed division algebras]]



\end{document}