nLab Hurwitz theorem

not to be confused with the Hurewicz theorem.

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Statement

The only Euclidean composition algebras / division composition algebras over the real numbers (i.e. real normed division algebras) are, up to isomorphism, the algebras of

1. real numbers

2. complex numbers

3. quaternions

4. octonions.

For more see at composition algebra – Hurwitz theorem.

Implications

Real spinor representations and supersymmetry

This classification turns out to closely connect to various other systems of exceptional structures in mathematics and physics. Notably 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 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 and geometry of physics – supersymmetry.

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,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

Hopf invariant one maps

The classification matches that of Hopf invariant one maps. See there for more.

See also at cross product.

Generalized Hurwitz Theorem

As describe in Elduque (2021), the Hurwitz Theorem can be extended to more general composition algebras that are not necessarily unital, but that are para-unital or even non-unital (see here for the relevant definitions).

The Generalized Hurwitz Theorem then becomes the statement that the only Euclidean composition algebras over the real numbers are, up to isomorphism:

1. Seven unital algebras: real numbers, complex numbers, quaternions, and octonions (along with their split companions).

2. Seven para-unital algebras, corresponding to the para- counterparts of the unital algebras above.

3. Two eight-dimensional non-unital algebras: the Okubo algebra? $\mathcal{O}$ and its split counterpart $\mathcal{O}_s$.

References

Due to Adolf Hurwitz (1859–1919), published posthumously in 1923.

• Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898) pp 309–316 (GDZ pdf, EuDML entry)

• Adolf Hurwitz, Über die Komposition der quadratischen Formen, Math. Ann. 88 (1923) pp 1–25, doi:10.1007/bf01448439 (GDZ pdf, EuDML entry)

• Wikipedia, Hurwitz theorem (composition algebras)

• Alberto Elduque, Composition algebras: “Algebra and Applications I: Non-associative Algebras and Categories”, Chapter 2, pp. 27-57, edited by Abdenacer Makhlouf, Sciences-Mathematics, ISTEWiley, London 2021.