{wiki}

A <vector field> with a <bilinear map> into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither <associative> nor <commutative>.

Examples: https://en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples
* <complex numbers>, i.e. <\R^2> with complex number multiplication
* <\R^3> with the <cross product>
* <quaternions>, i.e. <\R^4> with the quaternion multiplication


Algebra over a field

{c}
{wiki=Cayley–Dickson construction}

Constructs the <quaternions> from <complex numbers>, <octonions> from <quaternions>, and keeps doubling like this indefinitely.


Cayley-Dickson construction

{disambiguate=real division algebras}
{c}
{tag=Classification (mathematics)}
{wiki}

= Classification of associative real division algebras
{synonym}
{title2}

There are 3: <real numbers>, <complex numbers> and <quaternions>.

Notably, the <octonions> are not <associative>.


Frobenius theorem

Frobenius theorem (real division algebras)

{title2=8}
{wiki}

Unlike the <quaternions>, it is non-<associative>.


Octonion

{wiki}

= $SU(2)$
{synonym}
{title2}

https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)

<Double cover> of <SO(3)>.

<Isomorphic> to the <quaternions>.


Special unitary of degree 2

{title2=$\H$}
{title2=4}
{wiki}

Kind of extends the <complex numbers>.

Some facts that make them stand out:
* one of the only three real <associative> <division algebras> in addition to the <real numbers> and <complex numbers>, according to the <classification of associative real division algebras>
* the simplest non-<commutative> <division algebra>. Contrast for example with <complex numbers> where multiplication is <commutative>


Ciro Santilli @cirosantilli 37

 Incoming links: Quaternion