dg2test5 - Differential Geometry 2 Test 5 Let SO(3) be the...

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Unformatted text preview: Differential Geometry 2 Test 5 Let SO(3) be the Lie group of rotations on R3 and let so(3) ⊂ L(R3 ) be its Lie algebra. 1. Show that the linear map ζ : R3 → R3 lies in so(3) iff x, y ∈ R3 ⇒ ζx · y + x · ζy = 0. 2. Prove that the map Φ : R3 → so(3) : z → z × ( ) is a Lie algebra isomorphism that is SO(3)-equivariant for the standard action on R3 and the adjoint action on so(3). 3. Verify that the rule x, y ∈ R3 ⇒ (Φx |Φy ) := x · y defines an Ad-invariant inner product (a multiple of the Killing form; which multiple?) on so(3). 4. Check that the linear isomorphism so(3) → so(3)∗ : ζ → (ζ | ) is SO(3)- equivariant for the adjoint action and the coadjoint action. 5. Confirm that the coadjoint orbits of SO(3) are two-spheres (that is, all but one of them). Note: Here, x · y and x × y indicate the standard scalar product and vector product on R3 . 1 ...
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