Editing Clifford algebra (section)

== Structure of Clifford algebras ==
''In this section we assume that characteristic is not {{math|2}}, the vector space {{math|V}} is finite-dimensional and that the associated symmetric bilinear form of {{math|Q}} is nondegenerate.'' 

A [[central simple algebra]] over {{math|''K''}} is a matrix algebra over a (finite-dimensional) division algebra with center {{math|''K''}}. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.
* If {{math|''V''}} has even dimension then {{math|Cl(''V'', ''Q'')}} is a central simple algebra over&nbsp;{{math|''K''}}.
* If {{math|''V''}} has even dimension then the even subalgebra {{math|Cl{{sup|[0]}}(''V'', ''Q'')}} is a central simple algebra over a quadratic extension of {{math|''K''}} or a sum of two isomorphic central simple algebras over&nbsp;{{math|''K''}}.
* If {{math|''V''}} has odd dimension then {{math|Cl(''V'', ''Q'')}} is a central simple algebra over a quadratic extension of {{math|''K''}} or a sum of two isomorphic central simple algebras over&nbsp;{{math|''K''}}.
* If {{math|''V''}} has odd dimension then the even subalgebra {{math|Cl{{sup|[0]}}(''V'', ''Q'')}} is a central simple algebra over&nbsp;{{math|''K''}}.

The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that {{math|''U''}} has even dimension and a non-singular bilinear form with [[discriminant]] {{math|''d''}}, and suppose that {{math|''V''}} is another vector space with a quadratic form. The Clifford algebra of {{math|''U'' + ''V''}} is isomorphic to the tensor product of the Clifford algebras of {{math|''U''}} and {{math|(−1)<sup>dim(''U'')/2</sup>''dV''}}, which is the space {{math|''V''}} with its quadratic form multiplied by {{math|(−1)<sup>dim(''U'')/2</sup>''d''}}. Over the reals, this implies in particular that
<math display="block"> \operatorname{Cl}_{p+2,q}(\mathbf{R}) = \mathrm{M}_2(\mathbf{R})\otimes \operatorname{Cl}_{q,p}(\mathbf{R}) </math>
<math display="block"> \operatorname{Cl}_{p+1,q+1}(\mathbf{R}) = \mathrm{M}_2(\mathbf{R})\otimes \operatorname{Cl}_{p,q}(\mathbf{R}) </math>
<math display="block"> \operatorname{Cl}_{p,q+2}(\mathbf{R}) = \mathbf{H}\otimes \operatorname{Cl}_{q,p}(\mathbf{R}). </math>
These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras;  see the [[classification of Clifford algebras]].

Notably, the [[Morita equivalence]] class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends on only the signature {{math|(''p'' − ''q'') mod 8}}. This is an algebraic form of [[Bott periodicity]].