# nLab quaternion

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

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

The automorphism group of the quaternions 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)

Lorentzian spacetime dimensionspin groupnormed division algebrabrane 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 quaternionslittle string
$10 = 9+1$$Spin(9,1) \underset{some\,sense}{\simeq} SL(2,\mathbb{O})$$\mathbb{O}$ the octonionsheterotic/type II string

## References

A survey is in

• T. Y. Lam, Hamilton’s Quaternions (ps)