Editing Clifford algebra (section)

=== Antiautomorphisms ===
In addition to the automorphism {{math|''α''}}, there are two [[antiautomorphism]]s that play an important role in the analysis of Clifford algebras. Recall that the [[tensor algebra]] {{math|''T''(''V'')}} comes with an antiautomorphism that reverses the order in all products of vectors:
<math display="block">v_1\otimes v_2\otimes \cdots \otimes v_k \mapsto v_k\otimes \cdots \otimes v_2\otimes v_1.</math>
Since the ideal {{math|''I''<sub>''Q''</sub>}} is invariant under this reversal, this operation descends to an antiautomorphism of {{math|Cl(''V'', ''Q'')}} called the ''transpose'' or ''reversal'' operation, denoted by {{math|''x''<sup>t</sup>}}. The transpose is an antiautomorphism: {{math|1=(''xy'')<sup>t</sup> = ''y''<sup>t</sup> ''x''<sup>t</sup>}}. The transpose operation makes no use of the {{math|'''Z'''<sub>2</sub>}}-grading so we define a second antiautomorphism by composing {{math|''α''}} and the transpose. We call this operation ''Clifford conjugation'' denoted <math>\bar x</math>
<math display="block">\bar x = \alpha(x^\mathrm{t}) = \alpha(x)^\mathrm{t}.</math>
Of the two antiautomorphisms, the transpose is the more fundamental.{{efn|The opposite is true when using the alternate (−) sign convention for Clifford algebras: it is the conjugate that is more important. In general, the meanings of conjugation and transpose are interchanged when passing from one sign convention to the other. For example, in the convention used here the inverse of a vector is given by {{math|1=''v''<sup>−1</sup> = ''v''<sup>t</sup> / ''Q''(''v'')}} while in the (−) convention it is given by {{math|1=''v''<sup>−1</sup> = {{overline|''v''}} / ''Q''(''v'')}}.}}

Note that all of these operations are [[involution (mathematics)|involutions]]. One can show that they act as {{math|±1}} on elements that are pure in the {{math|'''Z'''}}-grading. In fact, all three operations depend on only the degree modulo&nbsp;{{math|4}}. That is, if {{math|''x''}} is pure with degree {{math|''k''}} then
<math display="block">\alpha(x) = \pm x \qquad x^\mathrm{t} = \pm x \qquad \bar x = \pm x</math>
where the signs are given by the following table:
: {| class=wikitable
 ! {{math|''k'' mod 4}}
 | {{math|0}} || {{math|1}} || {{math|2}} || {{math|3}} || …
|-
 ! <math>\alpha(x)\,</math>
 | {{math|+}} || {{math|−}} || {{math|+}} || {{math|−}} || {{math|(−1)<sup>''k''</sup>}}
|-
 ! <math>x^\mathrm{t}\,</math>
 | {{math|+}} || {{math|+}} || {{math|−}} || {{math|−}} || {{math|(−1)<sup>''k''(''k'' − 1)/2</sup>}}
|-
 ! <math>\bar x</math>
 | {{math|+}} || {{math|−}} || {{math|−}} || {{math|+}} || {{math|(−1)<sup>''k''(''k'' + 1)/2</sup>}}
|}