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by Ciro Santilli (@cirosantilli, 37)

Lie algebra of a isometry group

 ... Area of mathematics Geometry Differential geometry Lie group Important Lie group Isometry group
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We can almost reach the Lie algebra of any isometry group in a single go. For every X in the Lie algebra we must have:
∀v,w∈V,t∈R(etXv∣etXw)=(v∣w)
(1)
because etX has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Then:
dtd(etXv∣etXw)​​0​=0⟹(XetXv∣etXw)+(etXv∣XetXw)​0​=0⟹(Xv∣w)+(v∣Xw)=0
(2)
so we reach:
∀v,w∈V(Xv∣w)=−(v∣Xw)
(3)
With this relation, we can easily determine the Lie algebra of common isometries:
  • Lie algebra of O(n)
Bibliography:
  • An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) page 151

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