Graduate Do Dynkin Diagrams Only Describe Lie Algebras?

[phoenix95]

TL;DR

Dynkin Diagrams

I have two questions:
1. It is said that just by using the Dynkin diagrams one can recover the entire algebra of the Lie group. But this is JUST the Algebra and not some representation, correct?
2. SL(n+1) and SU(n+1) have the same Dynkin diagrams, that also means have the same algebra?

 

[fresh_42]

1. Yes. However, simple Lie algebras, those with a Dynkin diagram, have no center and the adjoint representation is thus an isomorphic image of the algebra. I.e. you get one faithful and irreducible representation for free.

2. Yes, the Dynkin diagrams $A_n$.

Here's an example ($G_2$) how to retrieve it:
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/

 

[soloenergy]

1. Yes, you are correct. The Dynkin diagrams only give information about the algebra of the Lie group, not its representations. To fully understand the group, one would need to also consider its representations.

2. Yes, that is correct. Since SL(n+1) and SU(n+1) have the same Dynkin diagrams, they also have the same algebra. However, they may have different representations since they are different groups.