Contents

Idea

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

Normed division algebra structure

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

${i}^{2}={j}^{2}={k}^{2}=ijk=-1,$i^2 = j^2 = k^2 = i j k = -1,

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

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

where given an $ℝ$-linear combination $\alpha =a1+bi+cj+dk$, we define the conjugate $\overline{\alpha }≔a1-bi-cj-dk$. A simple calculation yields

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

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

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

In this way $ℍ$ is a normed division algebra.

Modules and bimodules

We have canonical left and right module structures on ${ℍ}^{n}$, but as $ℍ$ 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 ${M}_{n}\left(ℍ\right)$ (using the left and right module structures on ${ℍ}^{n}$ respectively), and only left eigenvalues relate to the spectrum of the associated linear operator.

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

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

A survey is in

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

Revised on August 28, 2013 15:07:45 by Urs Schreiber (82.113.98.24)