# nLab quaternion

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A quaternion or Hamilton number is a kind of number similar to the complex numbers but with three instead of one square root of $(-1)$ adjoined, satisfying certain relations.

The quaternions form the largest associative normed division algebra, usually denoted $\mathbb{H}$ after William Rowan Hamilton (since $\mathbb{Q}$ is taken for the rational numbers).

## Properties

### Normed division algebra structure

The structure of $\mathbb{H}$ as an $\mathbb{R}$-algebra is given by a basis $\{1, i, j, k\}$ of the underlying vector space of $\mathbb{H}$, equipped with a multiplication table where $1$ is the identity element and otherwise uniquely specified by the equations

$i^2 = j^2 = k^2 = i j k = -1,$

and extended by $\mathbb{R}$-linearity to all of $\mathbb{H}$. The norm on $\mathbb{H}$ is given by

${\|\alpha\|}^2 = \alpha \widebar{\alpha}$

where given an $\mathbb{R}$-linear combination $\alpha = a 1 + b i + c j + d k$, we define the conjugate $\widebar{\alpha} \coloneqq a 1 - b i - c j - d k$. A simple calculation yields

${\|\alpha\|}^2 = a^2 + b^2 + c^2 + d^2$

whence for $\alpha \neq 0$, the multiplicative inverse is

$\alpha^{-1} = \frac1{{\|\alpha\|}^2} \widebar{\alpha}.$

In this way $\mathbb{H}$ is a normed division algebra.

### Modules and bimodules

We have canonical left and right module structures on $\mathbb{H}^n$, but as $\mathbb{H}$ is not commutative, if we want to talk about tensor products of modules, we need to consider bimodules. This also means that ordinary linear algebra as is used over a field is not quite the same when dealing with quaternions. For instance, one needs to distinguish between left and right eigenvalues of matrices in $M_n(\mathbb{H})$ (using the left and right module structures on $\mathbb{H}^n$ respectively), and only left eigenvalues relate to the spectrum of the associated linear operator.

Using the conjugation operation one can define an inner product $\langle q,p\rangle := \overline{q} p$ on $\mathbb{H}^n$ so that the corresponding orthogonal group is the compact symplectic group.

### Automorphisms

###### Proposition

The automorphism group of the quaternions, as a real algebra, is SO(3), acting canonically on their imaginary part (in generalization of how the product of complex numbers respects the complex conjugation action)

(e.g. Klimov-Zhuravlev, p. 85)

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,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

Review