# I Simply-connected, complex, simple Lie groups

1. Dec 29, 2016

### Rasalhague

I've been looking at John Baez's lecture notes "Lie Theory Through Examples". In the first chapter, he says Dynkin diagrams classify various types of object, including "simply-connected, complex, simple Lie groups." He discusses the An case in detail. But what are the simply-connected, complex, simple Lie groups associated with Bn and Dn? SO(n,C) is not simply connected. Spin(n) and PSL(n,C) are compact and, says Baez, "a complex simple Lie group is never compact."

Is there a good source online which lists all the simply-connected, complex, simple Lie groups and their maximal compact subgroups as Baez does for An?

I believe Cn is Sp(2n,C) with maximal compact subgroup USp(2n,C), the latter commonly written Sp(n).

2. Dec 29, 2016

### Staff: Mentor

If one refers to the simply-connected part, then the connected component of $1$ is meant. And I think $O(n)$ isn't simply connected, $SO(n)$ is. If $n$ denotes the number of simple roots, or knots in the Dynkin diagram, then

$A_n \triangleq SL(n+1)$
$B_n \triangleq O(2n+1) , SO(2n+1)$
$C_n \triangleq SP(2n)$
$D_n \triangleq O(2n) , SO(2n)$

As it's from SUNY I suppose it's o.k. to quote here without violating any copyrights:
https://www.math.stonybrook.edu/~kirillov/mat552/liegroups.pdf

3. Dec 30, 2016

4. Jan 4, 2017

### Rasalhague

I guess he must have meant "connected" rather than "simply connected" when he writes that Dynkin diagrams classify "a bunch of things" including: "simply connected complex simple Lie group" and "compact simply connected simple Lie groups".