[代數] 求助一個代數群的問題

[WasabiSushi (田馥甄Hebe)]

Let G be a connected and simply connected algebraic group over real field or complex field, and let H be a closed subgroup of G. Denote by H' the universal covering group of H, and then the inclusion i:H->G gives an inclusion d(i):h->g of their corresponding Lie algebras. Hence, d(i) is lifted to an injective homomorphism i':H'->G. Moreover, i*p=i' where p is the covering map from H' to H. Since i' is injective, it follows that p must be injective, and so p is isomorphism. Therefore H is isomorphic to
H' which is simply connected.

This shows that any closed subgroup of a simply connected algebraic group is simply connected.

However, set G=SL(n,C) and H=SO(n,C). Then G is simply connected, but H is not.

我想了很久, 沒有想出來自己的推理有甚麼問題. 請問為甚麼會出現矛盾呢? 求高人解答. 不勝感激!

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※ 編輯: WasabiSushi (180.154.50.218), 01/27/2015 00:35:27
→ kerwinhui   : H' -> G not injective, easier eg: S^1 -> C*
→ WasabiSushi : 抱歉, C*=C-{0}不是simply connected的.
→ kerwinhui   : Oops, try S^1 -> SU(2); z |-> diag(z,z^*)
→ kerwinhui   : Or: maximal torus in s.c. Lie group
→ WasabiSushi : Thanks. As to your example, S^1=SO(2)->SU(2) is
→ WasabiSushi : lifted to Spin(2)->SU(2). Do you mean that this
→ WasabiSushi : is not injective?
→ kerwinhui   : Spin(2)->SO(2) is z|->z^2, Spin(2) is not s.c.
→ kerwinhui   : Spin(2)->SU(2) has kernel ±1
→ WasabiSushi : Sorry,I'm a bit confused.The fundamental group
→ WasabiSushi : of S^1 is Z,so the kernal of the covering map is
→ WasabiSushi : Z instead of {1,-1}.Actually,Spin(2) is just
→ WasabiSushi : isomorphic to R*.But what makes me fell strange
→ WasabiSushi : is that the Lie algebra injection R->su(2)
→ WasabiSushi : given by x|->diag(ix,-ix) doesn't lift to a
→ WasabiSushi : group homomorphism.
→ kerwinhui   : No!  Spin(2) is the unique double cover of SO(2)
→ kerwinhui   : and exponentiating from algebra to group does
→ kerwinhui   : not preserve injection.
→ kerwinhui   : otherwise there would be no maximal tori, a huge
→ kerwinhui   : blow to much of Lie theory
→ WasabiSushi : Oh,you are right.Spin(2) is not the universal
→ WasabiSushi : cover.Yes,exponentiating does not preserve
→ WasabiSushi : injection. Thank you so much!