Real normed division algebra

Definition

A real normed division algebra is a finite-dimensional unital algebra $A$ over $\mathbb R$ with a positive-definite Euclidean norm $\lVert\cdot\rVert$ satisfying

\lVert xy\rVert=\lVert x\rVert\,\lVert y\rVert

for all $x,y\in A$ . Equivalently, $N(x)=\lVert x\rVert^2$ makes $A$ a composition algebra with positive-definite norm.

Why division follows

Multiplicativity implies that $xy=0$ with $x\ne0$ forces $y=0$ , and similarly on the other side. More concretely, the standard conjugation satisfies

x^*x=xx^*=\lVert x\rVert^2 1,

so every nonzero element has the two-sided inverse

x^{-1}=\frac{x^*}{\lVert x\rVert^2}.

Classification

By Hurwitz's theorem, every real normed division algebra is isomorphic to exactly one of

\mathbb R,\qquad \mathbb C,\qquad \mathbb H,\qquad \mathbb O,

of real dimensions $1,2,4,8$ . Associativity holds for the first three and fails for $\mathbb O$ , while all four are alternative.

Convention warning

The word division algebra is sometimes reserved for associative algebras or defined only by solvability of $ax=b$ and $ya=b$ . “Normed division algebra” in the Hurwitz setting includes nonassociative algebras, requires a unit and a multiplicative positive-definite norm, and is finite-dimensional here.

References

John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §§1–2.
John H. Conway and Derek A. Smith, On Quaternions and Octonions, A K Peters, 2003. DOI record. Relevant: Chapters 1–3.