A division algebra is a possibly non-associative algebra $A$, typically over a field $k$, with the property that $a b = 0$ implies either $a = 0$ or $b = 0$ (or $a b \ne 0$ 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 $\mathbb{R}$ (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 dimension | spin group | normed division algebra | brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\mathbb{R}$ the real numbers | |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\mathbb{C}$ the complex numbers | |
$6 = 5+1$ | $Spin(5,1) \simeq SL(2, \mathbb{H})$ | $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})$ | $\mathbb{O}$ the octonions | heterotic/type II string |
See at division algebra and supersymmetry.