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dg2test5 - Dierential Geometry 2 Test 5 Let SO(3 be the Lie...

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Differential Geometry 2 Test 5 Let SO (3) be the Lie group of rotations on R 3 and let so (3) L ( R 3 ) be its Lie algebra. 1. Show that the linear map ζ : R 3 R 3 lies in so (3) iff x, y R 3 ζx · y + x · ζy = 0 . 2. Prove that the map Φ : R 3 so (3) : z z × ( ) is a Lie algebra isomorphism that is SO (3)-equivariant for the standard action on R 3 and the adjoint action on so (3). 3. Verify that the rule x, y R 3 x | Φ y ) := x · y defines an Ad-invariant inner product (a multiple of the Killing form; which
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