nLab Hurwitz theorem

Contents

not to be confused with the Hurewicz theorem.

Context

Algebra

higher algebra

universal algebra

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

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 groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq SL(2, \mathbb{H})$$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}$$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string

Hopf invariant one maps

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