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Orthogonal Groups

  • Morton L. Curtis
Part of the Universitext book series (UTX)

Abstract

We have a consistent notion of conjugation for R ⊂ C ⊂ H. Namely,
$$ {\text{for x }} \in {\text{ R , }}\overline {\text{x}} {\text{ = x }}{\text{.}} $$
$$ {\text{For }}\alpha {\text{ = x + iy }} \in {\text{ C , }}\overline \alpha {\text{ = x - iy }}{\text{.}} $$
$$ {\text{For q = x + iy + jz + kw }} \in {\text{ H , }}\overline {\text{q}} {\text{ = x - iy - jz - kw }}{\text{.}} $$
We clearly have \( \overline{\overline \alpha } {\text{ = }}\alpha \) in all cases and
$$ (\overline {\alpha {\text{ + B}}} {\text{) = }}\overline \alpha {\text{ + }}\overline {\text{B}} \;. $$
It is an exercise to prove that
$$ \overline {\alpha {\text{B}}} \;{\text{ = }}\;\overline {{\text{B}}\alpha } \;. $$
Of course for R or C this is the same as
$$ \overline {\alpha {\text{B}}} \; = \;\overline {\alpha {\text{B}}} \;. $$

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Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Morton L. Curtis
    • 1
  1. 1.Department of MathematicsRice UniversityHoustonUSA

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