nLab Hurwitz theorem


not to be confused with the Hurewicz theorem.



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.


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

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/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? 𝒪\mathcal{O} and its split counterpart 𝒪 s\mathcal{O}_s.


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

Last revised on October 4, 2023 at 21:33:46. See the history of this page for a list of all contributions to it.