nLab
Hurwitz theorem

Contents

not to be confused with the Hurewicz theorem.


Contents

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
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)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)Spin(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.

References

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

Last revised on September 17, 2018 at 21:17:26. See the history of this page for a list of all contributions to it.