Editing Clifford algebra (section)

== Examples: real and complex Clifford algebras ==
The most important Clifford algebras are those over [[real number|real]] and [[complex number|complex]] vector spaces equipped with [[nondegenerate quadratic form]]s.

Each of the algebras {{math|Cl{{sub|''p'',''q''}}('''R''')}} and {{math|Cl{{sub|''n''}}('''C''')}} is isomorphic to {{math|''A''}} or {{math|''A'' ⊕ ''A''}}, where {{math|''A''}} is a [[Matrix ring|full matrix ring]] with entries from {{math|'''R'''}}, {{math|'''C'''}}, or&nbsp;{{math|'''H'''}}. For a complete classification of these algebras see ''[[Classification of Clifford algebras]]''.

=== Real numbers ===
{{main|Geometric algebra}}

Clifford algebras are also sometimes referred to as [[geometric algebra]]s, most often over the real numbers.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
<math display="block">Q(v) = v_1^2 + \dots + v_p^2 - v_{p+1}^2 - \dots - v_{p+q}^2 ,</math>
where {{math|1=''n'' = ''p'' + ''q''}} is the dimension of the vector space. The pair of integers {{math|(''p'', ''q'')}} is called the [[metric signature|signature]] of the quadratic form. The real vector space with this quadratic form is often denoted {{math|1='''R'''<sup>''p'',''q''</sup>.}} The Clifford algebra on {{math|1='''R'''{{sup|''p'',''q''}}}} is denoted {{math|1=Cl{{sub|''p'',''q''}}('''R''').}}  The symbol {{math|1=Cl{{sub|''n''}}('''R''')}}  means either {{math|1=Cl{{sub|''n'',0}}('''R''')}} or {{math|1=Cl{{sub|0,''n''}}('''R''')}}, depending on whether the author prefers positive-definite or negative-definite spaces.

A standard [[basis (linear algebra)|basis]] {{math|{{mset|''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>}}}} for {{math|1='''R'''<sup>''p'',''q''</sup>}} consists of {{math|1=''n'' = ''p'' + ''q''}} mutually orthogonal vectors, {{math|1=''p''}} of which square to {{math|+1}} and {{math|1=''q''}} of which square to&nbsp;{{math|−1}}. Of such a basis, the algebra {{math|1=Cl{{sub|''p'',''q''}}('''R''')}} will therefore have {{math|1=''p''}} vectors that square to {{math|+1}} and {{math|1=''q''}} vectors that square to&nbsp;{{math|−1}}.

A few low-dimensional cases are:
* {{math|1=Cl{{sub|0,0}}('''R''')}} is naturally isomorphic to {{math|1='''R'''}} since there are no nonzero vectors.
* {{math|1=Cl{{sub|0,1}}('''R''')}} is a two-dimensional algebra generated by {{math|1=''e''<sub>1</sub>}} that squares to {{math|−1}}, and is algebra-isomorphic to {{math|1='''C'''}}, the field of [[complex number]]s.
* {{math|1=Cl{{sub|1,0}}('''R''')}} is a two-dimensional algebra generated by {{math|1=''e''<sub>1</sub>}} that squares to {{math|1}}, and is algebra-isomorphic to the [[split-complex number]]s.
* {{math|1=Cl{{sub|0,2}}('''R''')}} is a four-dimensional algebra spanned by {{math|{{mset|1, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>1</sub>''e''<sub>2</sub>}}}}. The latter three elements all square to {{math|−1}} and anticommute, and so the algebra is isomorphic to the [[quaternion]]s&nbsp;{{math|1='''H'''}}.
* {{math|1=Cl{{sub|2,0}}('''R''') ≅ Cl{{sub|1,1}}('''R''')}} is isomorphic to the algebra of [[split-quaternion]]s.
* {{math|1=Cl{{sub|0,3}}('''R''')}} is an 8-dimensional algebra isomorphic to the [[Direct sum of modules#Direct sum of algebras|direct sum]] {{math|'''H''' ⊕ '''H'''}}, the [[split-biquaternion]]s.
* {{math|1=Cl{{sub|3,0}}('''R''') ≅ Cl{{sub|1,2}}('''R''')}}, also called the [[Pauli algebra]],{{sfn|Garling|2011|p=112}}{{sfn|Francis|Kosowsky|2005|p=404}} is isomorphic to the algebra of [[biquaternion]]s.

=== Complex numbers ===
One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension {{math|''n''}} is equivalent to the standard diagonal form
<math display="block">Q(z) = z_1^2 + z_2^2 + \dots + z_n^2.</math>
Thus, for each dimension {{math|''n''}}, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on {{math|'''C'''<sup>''n''</sup>}} with the standard quadratic form by {{math|Cl{{sub|''n''}}('''C''')}}.

For the first few cases one finds that
* {{math|Cl{{sub|0}}('''C''') ≅ '''C'''}}, the [[complex number]]s
* {{math|Cl{{sub|1}}('''C''') ≅ '''C''' ⊕ '''C'''}}, the [[bicomplex number]]s
* {{math|Cl{{sub|2}}('''C''') ≅ M<sub>2</sub>('''C''')}}, the [[biquaternion]]s
where {{math|M<sub>''n''</sub>('''C''')}} denotes the algebra of {{math|''n'' × ''n''}} matrices over {{math|'''C'''}}.