There is a close relationship between
the four normed division algebras
This is based on the fact that in certain dimensions, spin group representations are naturally identified with a , for one of the normed division algebras, see at spin group The exceptional isomorphisms.
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
|the real numbers|
|the complex numbers|
|the quaternions||little string|
|the octonions||heterotic/type II string|
The structure of the normed division algebras governs the existence of the brane scan and the super-∞-Lie algebras such as the supergravity Lie 3-algebra. By the D'Auria-Fre formulation of supergravity the ∞-Lie algebra valued forms with values in these constitute the field content of (11-dimensional) supergravity.
The relation was apparently first made explicit in
A pedagogical review is in
Discussion with an emphasis on the application to super Yang-Mills theory is in
Rafael I. Nepomechie, Nonabelian bosonization, triality, and superstring theory Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210
I. Bars, D. Nemschansky and S. Yankielowicz, SLACPub- 3758.
H. Tachibana, K. Imeda, Octonions, superstrings and ten-dimensional spinors , Il nuovo cimento, Vol 104 B N.1
The relation of the division algebras to ordinary (Lie algebraic) extensions of the super Poincare Lie algebra is discussed in
Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory (pdf)