Simply-connected, complex, simple Lie groups

[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 A_n case in detail. But what are the simply-connected, complex, simple Lie groups associated with B_n and D_n? 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 A_n?

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

 

[fresh_42]

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

 

[Rasalhague]

SO(2,C) is connected, but not simply connected (Fulton & Harris: Representation Theory, Prop. 23.1).

https://en.wikipedia.org/wiki/Orthogonal_group#Over_the_complex_number_field

 

[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".