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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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- Spring '09