Editing Clifford algebra (section)

=== Grading ===
In the following, assume that the characteristic is not&nbsp;{{math|2}}.{{efn|Thus the [[group ring|group algebra]] {{math|''K''['''Z'''{{px2}}/{{px2}}2'''Z''']}} is [[Semisimple algebra|semisimple]] and the Clifford algebra splits into eigenspaces of the main involution.}}

Clifford algebras are {{math|'''Z'''<sub>2</sub>}}-[[graded algebra]]s (also known as [[superalgebra]]s).  Indeed, the linear map on {{math|''V''}} defined by {{math|''v'' ↦ −''v''}} ([[reflection through the origin]]) preserves the quadratic form {{math|''Q''}} and so by the universal property of Clifford algebras extends to an algebra [[automorphism]]
<math display="block">\alpha: \operatorname{Cl}(V, Q) \to \operatorname{Cl}(V, Q).</math>

Since {{math|''α''}} is an [[Involution (mathematics)|involution]] (i.e. it squares to the [[identity function|identity]]) one can decompose {{math|Cl(''V'', ''Q'')}} into positive and negative eigenspaces of&nbsp;{{math|''α''}}
<math display="block">\operatorname{Cl}(V, Q) = \operatorname{Cl}^{[0]}(V, Q) \oplus \operatorname{Cl}^{[1]}(V, Q)</math>
where 
<math display="block">\operatorname{Cl}^{[i]}(V, Q) = \left\{ x \in \operatorname{Cl}(V, Q) \mid \alpha(x) = (-1)^i x \right\}.</math>

Since {{math|''α''}} is an automorphism it follows that:
<math display="block">\operatorname{Cl}^{[i]}(V, Q)\operatorname{Cl}^{[j]}(V, Q) = \operatorname{Cl}^{[i+j]}(V, Q)</math>
where the bracketed superscripts are read modulo&nbsp;2. This gives {{math|Cl(''V'', ''Q'')}} the structure of a {{math|'''Z'''<sub>2</sub>}}-[[graded algebra]].  The subspace {{math|Cl{{sup|[0]}}(''V'', ''Q'')}} forms a [[subalgebra]] of {{math|Cl(''V'', ''Q'')}}, called the ''even subalgebra''. The subspace {{math|Cl{{sup|[1]}}(''V'', ''Q'')}} is called the ''odd part'' of {{math|Cl(''V'', ''Q'')}} (it is not a subalgebra).  {{math|This '''Z'''<sub>2</sub>}}-grading plays an important role in the analysis and application of Clifford algebras. The automorphism {{math|''α''}} is called the ''main [[involution (mathematics)|involution]]'' or ''grade involution''. Elements that are pure in this {{math|'''Z'''<sub>2</sub>}}-grading are simply said to be even or odd.

''Remark''. The Clifford algebra is not a  {{math|'''Z'''}}-graded algebra, but is {{math|'''Z'''}}-[[filtered algebra|filtered]], where {{math|Cl{{sup|≤''i''}}(''V'', ''Q'')}} is the subspace spanned by all products of at most {{math|''i''}} elements of&nbsp;{{math|'''V'''}}.
<math display="block">\operatorname{Cl}^{\leqslant i}(V, Q) \cdot \operatorname{Cl}^{\leqslant j}(V, Q) \subset \operatorname{Cl}^{\leqslant i+j}(V, Q).</math>

The ''degree'' of a Clifford number usually refers to the degree in the {{math|'''Z'''}}-grading.

The even subalgebra {{math|Cl{{sup|[0]}}(''V'', ''Q'')}} of a Clifford algebra is itself isomorphic to a Clifford algebra.{{efn|Technically, it does not have the full structure of a Clifford algebra without a designated vector subspace, and so is isomorphic as an algebra, but not as a Clifford algebra.}}{{efn|We are still assuming that the characteristic is not {{math|2}}.}} If {{math|''V''}} is the [[orthogonal direct sum]] of a vector {{math|''a''}} of nonzero norm {{math|''Q''(''a'')}} and a subspace {{math|''U''}}, then {{math|Cl{{sup|[0]}}(''V'', ''Q'')}} is isomorphic to {{math|Cl(''U'', −''Q''(''a'')''Q''{{!}}{{smallsub|''U''}})}}, where {{math|''Q''{{!}}{{smallsub|''U''}}}} is the form {{math|''Q''}} restricted to {{math|''U''}}. In particular over the reals this implies that:
<math display="block">\operatorname{Cl}_{p,q}^{[0]}(\mathbf{R}) \cong \begin{cases}
  \operatorname{Cl}_{p,q-1}(\mathbf{R}) & q > 0 \\
  \operatorname{Cl}_{q,p-1}(\mathbf{R}) & p > 0
\end{cases}</math>

In the negative-definite case this gives an inclusion {{math|Cl{{sub|0,''n'' − 1}}('''R''') ⊂ Cl{{sub|0,''n''}}('''R''')}}, which extends the sequence
{{block indent|em=1.5|text={{math|'''R''' ⊂ '''C''' ⊂ '''H''' ⊂ '''H''' ⊕ '''H''' ⊂ ⋯}}}}

Likewise, in the complex case, one can show that the even subalgebra of {{math|Cl{{sub|''n''}}('''C''')}} is isomorphic to {{math|Cl{{sub|''n''−1}}('''C''')}}.