Lie Algebras of compact lie groups

Question

Can we know that a lie group is compact by studying its lie algebra ? If we consider the exponential function from the lie algebra to the corresponding lie group , we expect that exponentiating linear combinations of the generators in which the coefficients are extremely large gives rise to group elements extremely far away from the identity. So , how can exponentiating vectors whose norm is extremely large gives rise to group elements near the identity ?

Answer 1

Suppose that the Lie algebra ${\mathfrak g}$ of your (connected!) Lie group $G$ has zero center ${\mathfrak g}_0$. Then the necessary and sufficient condition for $G$ to be compact is that the Killing form of ${\mathfrak g}$ is negative definite. This result will be in any textbook on Lie theory. Start here. If the center ${\mathfrak g}_0$ is nonzero, you cannot conclude anything and need further information about the fundamental group of your Lie group.