Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The quaternions form the largest associative normed division algebra, usually denoted after William Rowan Hamilton (since is taken for the rational numbers).
Normed division algebra structure
The structure of as an -algebra is given by a basis of the underlying vector space of , equipped with a multiplication table where is the identity element and otherwise uniquely specified by the equations
and extended by -linearity to all of . The norm on is given by
where given an -linear combination , we define the conjugate . A simple calculation yields
whence for , the multiplicative inverse is
In this way is a normed division algebra.
Modules and bimodules
We have canonical left and right module structures on , but as 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 (using the left and right module structures on respectively), and only left eigenvalues relate to the spectrum of the associated linear operator.
Using the conjugation operation one can define an inner product on 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|
| the real numbers|
| the complex numbers|
| the quaternions||little string|
| the octonions||heterotic/type II string|
A survey is in
- T. Y. Lam, Hamilton’s Quaternions (ps)
- D.M. Klimov, V. Ph. Zhuravlev, Group-Theoretic Methods in Mechanics and Applied Mathematics