symmetric monoidal (∞,1)-category of spectra
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).
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
and extended by $\mathbb{R}$-linearity to all of $\mathbb{H}$. The norm on $\mathbb{H}$ is given by
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
whence for $\alpha \neq 0$, the multiplicative inverse is
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.
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)
See also at normed division algebra – automorphism
(e.g. Klimov-Zhuravlev, p. 85)
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 |
$10 = 9+1$ | $Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}$ | $\mathbb{O}$ the octonions | heterotic/type II string |
A survey is in
See also