not to be confused with the Hurewicz theorem.
symmetric monoidal (∞,1)-category of spectra
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.
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 |
$6 = 5+1$ | $Spin(5,1) \simeq SL(2, \mathbb{H})$ | $\phantom{A}$ $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) {\simeq}$ “SL(2,O)” | $\phantom{A}$ $\mathbb{O}$ the octonions | heterotic/type II string |
The classification matches that of Hopf invariant one maps. See there for more.
See also at cross product.
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)
Last revised on September 17, 2018 at 21:17:26. See the history of this page for a list of all contributions to it.