nLab quaternion

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)

See also at normed division algebra – automorphism

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

Related concepts

• quaternionic vector space

• quaternionic vector bundle

• Dieudonné determinant

• SL(2,H)

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

• supersymmetry and division algebra

• quaternionic quantum mechanics

• quaternion group, quaternionic unitary group

• tessarines

References

Monographs:

• D. M. Klimov, V. Ph. Zhuravlev, Group-Theoretic Methods in Mechanics and Applied Mathematics, Routledge (2004, 2020) [ISBN:9780367446987]

• Ernst Binz, Sonja Pods, Ch 1 in: The geometry of Heisenberg groups — With Applications in Signal Theory, Optics, Quantization, and Field Quantization, Mathematical Surveys and Monographs 151, American Mathematical Society (2008) [ams:surv-151]

• Tevian Dray, Corinne Manogue, Section 3.1 of: The Geometry of Octonions, World Scientific (2015) [doi:10.1142/8456]

See also:

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