Differential structure of the group of automorphism of a Lie group

In summary, the paper discusses the differential structure of the group of automorphisms of a Lie group, exploring how these automorphisms can be understood as smooth transformations that preserve the group’s structure. It analyzes the properties of the automorphism group, including its topology and differentiable structure, and provides insights into the relationship between the Lie algebra of the group and the corresponding automorphism group. The findings highlight the significance of these structures in understanding the symmetries and dynamics of Lie groups in mathematical and physical contexts.
  • #1
padawan
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I am working on this

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I am having trouble with b and c:

b) Suppose ##(f_n)_{n=1}^{\infty}## is a sequence in ##Aut(G)##, such that ##(T_e(f_n))_{n=1}^{\infty} \to \psi## converges in ##Aut(\mathfrak g)##

I want to show that ## f := \lim_{n\to \infty} f_n## exists as an continuous automorphism of the abstract group ##G##

First of all the ##f_n## are smooth, because they are in Aut(G), so they are Lie group isomorphisms, and therefore smooth by definition

Then by a known property of the exp:

##f_n(\exp(X))=\exp(T_ef_n X), \forall X \in T_eG## taking the limit yields:

##\lim_{n\to \infty} f_n(\exp(X))=\exp(\psi(X))\in G, \forall X \in T_eG##

##\lim_{n\to \infty} f_n(g)=\exp(\psi(X))=:f(g)\in G, \forall g## in a nbhd of the identity

I know that the image of the exponential map generates the connected component of the identity ##G_e##,and by connectedness this coincides with G so:
##G=G_e=<(T_eG)>=<\exp(\psi(T_eG))>=<\exp(T_eG)>##

This means that convergence is actually valid in the whole group, becuase if I have convergence in the generating set, I must have convergence in the generated set.

I hope this is correct,if not please tell me. Still I have to prove that it is bijective and a group homomorphism. I am not sure how to do this

b) How do I argue from here that ##f## is continuous? First I thought it was automatic, but then I recalled from analysis that pointwise convergence of a sequence of functions does not imply continuity. So I am clueless about how to proceed here as well
 
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