# nLab division algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A division algebra is a possibly non-associative algebra $A$, typically over a field $k$, which is also a division ring, hence with the property that for any $a,b \in A$ then $a b = 0$ 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 $k$), 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 many applications (also to physics) the most interesting division algebras are the normed division algebras over the real numbers: By the Hurwitz theorem these are the real numbers, complex numbers, quaternions and octonions. These have important relations to 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}$SL(2,O)$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string

## References

Last revised on December 2, 2018 at 10:15:06. See the history of this page for a list of all contributions to it.