nLab
quaternion

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).

Normed division algebra structure

Concretely, the structure of \mathbb{H} as an \mathbb{R}-algebra is given by a basis {1,i,j,k}\{1, i, j, k\} of the underlying vector space of \mathbb{H}, equipped with a multiplication table where 11 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 \mathbb{R}-linearity to all of \mathbb{H}. The norm on \mathbb{H} is given by

α 2=αα¯{\|\alpha\|}^2 = \alpha \widebar{\alpha}

where given an \mathbb{R}-linear combination α=a1+bi+cj+dk\alpha = a 1 + b i + c j + d k, we define the conjugate α¯a1bicjdk\widebar{\alpha} \coloneqq a 1 - b i - c j - d k. A simple calculation yields

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

whence for α0\alpha \neq 0, the multiplicative inverse is

α 1=1α 2α¯.\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 n\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()M_n(\mathbb{H}) (using the left and right module structures on n\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 q,p:=q¯p\langle q,p\rangle := \overline{q} p on n\mathbb{H}^n so that the corresponding orthogonal group is the compact symplectic group.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) somesenseSL(2,𝕆)Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})𝕆\mathbb{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)