# the map exp for compact real lie groups

Something I'm reading says that for a compact connected complex Lie group $G$, the kernel of $\exp:T_e(G) \to G$ is a lattice $\Lambda$ in $T_e(G)$, and $\exp$ is surjective, so $G \simeq T_e(G)/\Lambda$. What is the story for real compact connected Lie groups? Looking at the classification of compact Lie groups, get things besides tori, e.g. $SO(n)$ and $SU(n)$. In these cases, is $\exp$ injective, surjective, or is the kernel a lattice and is the image a maximal torus?

• Have you tried looking in the standard textbooks on this subject or even googling? This is a question that comes up extremely often, even in this site. – Mariano Suárez-Álvarez Jun 3 '17 at 16:39
• I apologize! I didn't look it up myself, because I'm just experimenting with if asking questions on mathstackexchange can help me as I read one thing. I know i need to learn the fundamentals of Lie groups so again I apologize – usr0192 Jun 3 '17 at 16:48
• The map $\exp$ is not a homomorphism, unless G is abelian, so knowing its "kernel" is not very useful. I suggest you work out the case of G=SO(3). – Moishe Kohan Jun 3 '17 at 17:18
• math.stackexchange.com/questions/1089636/… – Moishe Kohan Jun 3 '17 at 17:20