Hello,
i'm new here on the bord. I hope i don't do anything wrong. I have problems with an exercise about Lie Groups, and their corresponding Lie Algebra:
So Let G be a Lie Group, \(\displaystyle A=T_eG\) the corresponding Lie Algebra and \(\displaystyle X \in A\). I want to show this equation:
\(\displaystyle \phi^{L_X}_t(h)=R_{\phi^{L_X}_t(1)}(h)\), where\(\displaystyle \phi^{L_X}_t\) is the flow corresponding to the vectorfield \(\displaystyle L_X\), i.e. \(\displaystyle \phi^{L_X}_t=c_h(t)\), where \(\displaystyle c_h(t)\) is the integral curve of \(\displaystyle L_X\) with starting point h. The R on the RHS is the right multiplication. I hope now everything is clear (if not ask me)
So i have write the equation to another form:
\(\displaystyle \phi^{L_X}_t(h)=R_{\phi^{L_X}_t(1)}(h)\) <=> \(\displaystyle c_h(t)=h*c_1(t)\), where \(\displaystyle c_h, c_1\) are integral cures to the vf \(\displaystyle L_X\)with starting point h, 1 respectively.
If i take t=0, then of course the equation becomes correct, since \(\displaystyle c_h(0)=h=h*1=h*c_1(0)\). But why is this equation right for all \(\displaystyle t \in \mathbb{R}\)??
I hope you can help me.
Regards
i'm new here on the bord. I hope i don't do anything wrong. I have problems with an exercise about Lie Groups, and their corresponding Lie Algebra:
So Let G be a Lie Group, \(\displaystyle A=T_eG\) the corresponding Lie Algebra and \(\displaystyle X \in A\). I want to show this equation:
\(\displaystyle \phi^{L_X}_t(h)=R_{\phi^{L_X}_t(1)}(h)\), where\(\displaystyle \phi^{L_X}_t\) is the flow corresponding to the vectorfield \(\displaystyle L_X\), i.e. \(\displaystyle \phi^{L_X}_t=c_h(t)\), where \(\displaystyle c_h(t)\) is the integral curve of \(\displaystyle L_X\) with starting point h. The R on the RHS is the right multiplication. I hope now everything is clear (if not ask me)
So i have write the equation to another form:
\(\displaystyle \phi^{L_X}_t(h)=R_{\phi^{L_X}_t(1)}(h)\) <=> \(\displaystyle c_h(t)=h*c_1(t)\), where \(\displaystyle c_h, c_1\) are integral cures to the vf \(\displaystyle L_X\)with starting point h, 1 respectively.
If i take t=0, then of course the equation becomes correct, since \(\displaystyle c_h(0)=h=h*1=h*c_1(0)\). But why is this equation right for all \(\displaystyle t \in \mathbb{R}\)??
I hope you can help me.
Regards
