- #1
Calabi
- 140
- 2
Hello, let be ##G## a connected Lie group. I suppose##Ad(G) \subset Gl(T_{e}G)## is compact and the center ## Z(G)## of ##G## is discret (just to remember, forall ##g \in G##, ##Ad(g) = T_{e}i_{g}## with ##i_{g} : x \rightarrow gxg^{-1}##.).
I saw without any proof that in those hypothesis ##G## is compact and the center is finite.
Have you got a proof of this results please?
Here is the things I try to do : if we show ##G## compact, then since the center is closed and discret then it will be finite. But since ##G## is connected we show that ##Z(G)## is the kernel of
##Ad## so as ##Ad(G)## is compact it's also a Lie group so that ##Ad(G)## is isomorph to ##G / Z(G)##. Then if I show the commutator group is dense in ##G / Z(G)##, I saw in a books that we get the result but I don't know how to do.
Thank you in advance and have a nice afternoon.
I saw without any proof that in those hypothesis ##G## is compact and the center is finite.
Have you got a proof of this results please?
Here is the things I try to do : if we show ##G## compact, then since the center is closed and discret then it will be finite. But since ##G## is connected we show that ##Z(G)## is the kernel of
##Ad## so as ##Ad(G)## is compact it's also a Lie group so that ##Ad(G)## is isomorph to ##G / Z(G)##. Then if I show the commutator group is dense in ##G / Z(G)##, I saw in a books that we get the result but I don't know how to do.
Thank you in advance and have a nice afternoon.