# Contents

## Definition

A division algebra is a possibly non-associative algebra $A$, typically over a field $k$, with the property that $ab=0$ implies either $a=0$ or $b=0$ (or $ab\ne 0$ whenever $a,b\ne 0$). If $A$ is finite-dimensional (over a field), this is equivalent to assuming that for any nonzero $a\in A$, the operations of left multiplication and right multiplication by $a$ are invertible. If furthermore $A$ is also associative and unital, this is also equivalent to the existence, for each nonzero $a\in A$, of an element ${a}^{-1}\in A$ with $a{a}^{-1}={a}^{-1}a=1$. However, it is easy to construct nonassociative unital finite-dimensional algebras over $ℝ$ (the field of real numbers) such that either:

• $A$ is not a division algebra but for each nonzero $a\in A$ there exists ${a}^{-1}\in A$ with $a{a}^{-1}={a}^{-1}a=1$.

• $A$ is a division algebra but there exists nonzero $a\in A$ for which there is no ${a}^{-1}\in A$ with $a{a}^{-1}={a}^{-1}a=1$.

For applications to physics, the most interesting division algebras are probably the normed division algebras: the real numbers, complex numbers, quaternions and octonions. These have important relations to supersymmetry.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
$3=2+1$$\mathrm{Spin}\left(2,1\right)\simeq \mathrm{SL}\left(2,ℝ\right)$$ℝ$ the real numbers
$4=3+1$$\mathrm{Spin}\left(3,1\right)\simeq \mathrm{SL}\left(2,ℂ\right)$$ℂ$ the complex numbers
$6=5+1$$\mathrm{Spin}\left(5,1\right)\simeq \mathrm{SL}\left(2,ℍ\right)$$ℍ$ the quaternionslittle string
$10=9+1$$\mathrm{Spin}\left(9,1\right){\simeq }_{\mathrm{some}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sense}}\mathrm{SL}\left(2,𝕆\right)$$𝕆$ the octonionsheterotic/type II string

## References

Revised on November 14, 2013 11:32:41 by Urs Schreiber (188.200.54.65)