Editing Hypercomplex number (section)

=== Clifford algebras ===
A [[Clifford algebra]] is the unital associative algebra generated over an underlying vector space equipped with a [[quadratic form]]. Over the real numbers this is equivalent to being able to define a symmetric scalar product, {{nowrap|1=''u'' ⋅ ''v'' = {{sfrac|1|2}}(''uv'' + ''vu'')}} that can be used to [[orthogonalization|orthogonalise]] the quadratic form, to give a basis {{nowrap|{{mset|''e''<sub>1</sub>, ..., ''e''<sub>''k''</sub>}}}} such that:
<math display="block">\frac{1}{2} \left(e_i e_j + e_j e_i\right) = \begin{cases}
  -1, 0, +1 & i = j, \\
          0 & i \not = j.
\end{cases}</math>

Imposing closure under multiplication generates a multivector space spanned by a basis of 2<sup>''k''</sup> elements, {{mset|1, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ..., ''e''<sub>1</sub>''e''<sub>2</sub>, ..., ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>, ...}}.  These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {{mset|''e''<sub>1</sub>, ..., ''e''<sub>''k''</sub>}}, the remaining basis elements need not [[Anticommutative property|anti-commute]], depending on how many simple exchanges must be carried out to swap the two factors.  So {{nowrap|1=''e''<sub>1</sub>''e''<sub>2</sub> = −''e''<sub>2</sub>''e''<sub>1</sub>}}, but {{nowrap|1=''e''<sub>1</sub>(''e''<sub>2</sub>''e''<sub>3</sub>) = +(''e''<sub>2</sub>''e''<sub>3</sub>)''e''<sub>1</sub>}}.

Putting aside the bases which contain an element ''e''<sub>''i''</sub> such that {{nowrap|1=''e''<sub>''i''</sub><sup>2</sup> = 0}} (i.e. directions in the original space over which the quadratic form was [[degenerate form|degenerate]]), the remaining Clifford algebras can be identified by the label Cl<sub>''p'',''q''</sub>(<math>\mathbb{R}</math>), indicating that the algebra is constructed from ''p'' simple basis elements with {{nowrap|1=''e''<sub>''i''</sub><sup>2</sup> = +1}}, ''q'' with {{nowrap|1=''e''<sub>''i''</sub><sup>2</sup> = −1}}, and where <math>\mathbb{R}</math> indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

These algebras, called [[geometric algebra]]s, form a systematic set, which turn out to be very useful in physics problems which involve [[rotation]]s, [[phase (waves)|phase]]s, or [[Spin (physics)|spin]]s, notably in [[classical mechanics|classical]] and [[quantum mechanics]], [[electromagnetic theory]] and [[theory of relativity|relativity]].

Examples include: the [[complex number]]s Cl<sub>0,1</sub>(<math>\mathbb{R}</math>), [[split-complex number]]s Cl<sub>1,0</sub>(<math>\mathbb{R}</math>), [[quaternion]]s Cl<sub>0,2</sub>(<math>\mathbb{R}</math>), [[split-biquaternion]]s Cl<sub>0,3</sub>(<math>\mathbb{R}</math>), [[split-quaternion]]s {{nowrap|Cl<sub>1,1</sub>(<math>\mathbb{R}</math>) ≈ Cl<sub>2,0</sub>(<math>\mathbb{R}</math>)}} (the natural algebra of two-dimensional space); Cl<sub>3,0</sub>(<math>\mathbb{R}</math>) (the natural algebra of three-dimensional space, and the algebra of the [[Pauli matrices]]); and the [[spacetime algebra]] Cl<sub>1,3</sub>(<math>\mathbb{R}</math>).

The elements of the algebra Cl<sub>''p'',''q''</sub>(<math>\mathbb{R}</math>) form an even subalgebra Cl{{su|lh=1em|p=[0]|b=''q''+1,''p''}}(<math>\mathbb{R}</math>) of the algebra Cl<sub>''q''+1,''p''</sub>(<math>\mathbb{R}</math>), which can be used to parametrise rotations in the larger algebra.  There is thus a close connection between complex numbers and rotations in two-dimensional space; between quaternions and rotations in three-dimensional space; between split-complex numbers and (hyperbolic) rotations ([[Lorentz transformations]]) in 1+1-dimensional space, and so on.

Whereas Cayley–Dickson and split-complex constructs with eight or more dimensions are not associative with respect to multiplication, Clifford algebras retain associativity at any number of dimensions.

In 1995 [[Ian R. Porteous]] wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:<ref>{{citation |author-link=Ian R. Porteous |first=Ian R. |last=Porteous |title=Clifford Algebras and the Classical Groups |publisher=[[Cambridge University Press]] |year=1995 |isbn=0-521-55177-3 |pages=88–89 }}</ref>
: Let ''A'' be a real associative algebra with unit element 1. Then
:* 1 generates <math>\mathbb{R}</math> ([[real number|algebra of real numbers]]),
:* any two-dimensional subalgebra generated by an element ''e''<sub>0</sub> of ''A'' such that {{nowrap|1=''e''<sub>0</sub><sup>2</sup> = −1}} is isomorphic to <math>\mathbb{C}</math> ([[complex number|algebra of complex number]]s),
:* any two-dimensional subalgebra generated by an element ''e''<sub>0</sub> of ''A'' such that {{nowrap|1=''e''<sub>0</sub><sup>2</sup> = 1}} is isomorphic to <math>\mathbb{R}</math><sup>2</sup> (pairs of real numbers with component-wise product, isomorphic to the [[split-complex number|algebra of split-complex numbers]]),
:* any four-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = -1</math> is isomorphic to <math>\mathbb{H}</math> ([[quaternion|algebra of quaternions]]),
:* any four-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = 1</math> is isomorphic to M<sub>2</sub>(<math>\mathbb{R}</math>) (2 × 2 [[real matrices]], [[coquaternion]]s),
:* any eight-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>, ''e''<sub>2</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = e_2 ^2 = -1</math> is isomorphic to <sup>2</sup><math>\mathbb{H}</math> ([[split-biquaternion]]s),
:* any eight-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>, ''e''<sub>2</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = e_2 ^2 = 1</math> is isomorphic to M<sub>2</sub>(<math>\mathbb{C}</math>) ({{nowrap|2 × 2}} complex matrices, [[biquaternion]]s, [[Pauli algebra]]).

{{for|extension beyond the classical algebras|Classification of Clifford algebras}}