Composition algebra - Knowlpedia

Definition

Let $k$ be a field of characteristic different from $2$ . A composition algebra over $k$ is a finite-dimensional unital algebra $A$ with a nondegenerate quadratic form $N:A\to k$ such that

N(xy)=N(x)N(y)

for all $x,y\in A$ . Here “composition algebra” means unital composition algebra, also called a Hurwitz algebra.

Trace and conjugation

Polarizing $N$ gives the symmetric bilinear form

\langle x,y\rangle=N(x+y)-N(x)-N(y).

After the normalization $N(1)=1$ , define the trace $t(x)=\langle x,1\rangle$ and the standard conjugation

x^*=t(x)1-x.

Every element satisfies

x^*x=xx^*=N(x)1, \qquad x^2-t(x)x+N(x)1=0.

If $N(x)\ne0$ , then $x^{-1}=x^*/N(x)$ .

Structure

Every composition algebra is alternative, and its dimension is $1$ , $2$ , $4$ , or $8$ . Over a general field the norm may be isotropic, producing zero divisors; thus a composition algebra need not be a division algebra. Over $\mathbb R$ , positive-definite norm gives $\mathbb R$ , $\mathbb C$ , $\mathbb H$ , or $\mathbb O$ , while indefinite forms give split composition algebras.

Convention warning

Some authors allow a composition algebra without a multiplicative identity. This broader class includes symmetric composition algebras such as Okubo algebras. The unital convention used here is the one relevant to Hurwitz's theorem and the classical real normed division algebras.

References

Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. DOI record. Relevant: Chapter 1.
Richard D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, 1966. Project Gutenberg edition. Relevant: Chapter III, §4.