# Simply-connected, complex, simple Lie groups

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• Rasalhague
In summary, John Baez's lecture notes "Lie Theory Through Examples" discuss how Dynkin diagrams classify various types of objects, including simply-connected, complex, simple Lie groups. While the An case is discussed in detail, there is no clear mention of the simply-connected, complex, simple Lie groups associated with Bn and Dn. Baez does mention that SO(n,C) is not simply connected and that a complex simple Lie group is never compact. There is a good online source that lists the simply-connected, complex, simple Lie groups and their maximal compact subgroups, similar to how Baez does for An. It is mentioned that Cn is equivalent to Sp(2n,C) with a maximal compact subgroup of
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).

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

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".

## 1. What is a simply-connected Lie group?

A simply-connected Lie group is a type of group in mathematics that is connected, meaning all of its elements can be continuously connected to each other, and is simply-connected, meaning it has no "holes" or "handles" in its structure. In other words, any closed loop in the group can be continuously shrunk to a single point without leaving the group. Simply-connected Lie groups are important in the study of geometry and topology.

## 2. What is the significance of being a complex Lie group?

A complex Lie group is a Lie group whose elements are defined by complex numbers. These types of groups have a rich structure and have been used extensively in physics, particularly in the study of symmetries in quantum mechanics. In addition, complex Lie groups have applications in algebraic geometry and algebraic topology.

## 3. How does a Lie group differ from other types of groups?

A Lie group is a type of continuous group, meaning its elements can be smoothly varied. This is in contrast to discrete groups, where the elements can only take on a finite number of values. Lie groups also have a smooth manifold structure, meaning they are locally similar to Euclidean space. This allows for the use of calculus and other tools from differential geometry in the study of Lie groups.

## 4. What does it mean for a Lie group to be simple?

A simple Lie group is a type of Lie group that cannot be broken down into smaller, non-trivial groups. In other words, it has no non-trivial normal subgroups. These groups are important in group theory and have applications in fields such as physics and chemistry.

## 5. How are Lie groups related to Lie algebras?

Lie groups and Lie algebras are closely related mathematical structures. A Lie algebra is a vector space equipped with a bilinear operation called the Lie bracket, which satisfies certain properties. Lie groups can be associated with Lie algebras through a process called Lie algebraic homomorphism, which allows for the study of Lie groups using the techniques of linear algebra. In fact, many properties of Lie groups can be understood in terms of their corresponding Lie algebras, making the study of both structures important in mathematics and physics.

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