quaternion

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).

Concretely, 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.

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.

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 |

A survey is in

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

Revised on March 16, 2015 14:09:38
by Urs Schreiber
(88.128.80.39)