4
$\begingroup$

My question is: How can I construct the Dynkin diagrams of a semisimple Lie algebra $L$ which is the direct sum of simple Lie algebras, such as for example $\text {su}(2)\oplus\text{su}(2)\oplus\text{su}(2)$? Is it the combination of Dynkin diagrams of the simple Lie algebras?

$\endgroup$
1
  • 2
    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Oct 6, 2013 at 18:34

2 Answers 2

3
$\begingroup$

From what I understand, for semisimple Lie algebra you draw disconnected Dynkin diagram. So for your example it would be three disconnected vertices (since the corresponding root system is $A_1\times A_1\times A_1$).

$\endgroup$
2
$\begingroup$

The Dynkin diagram displays information about the angles between the simple roots. If two simple roots are separated by 90 degrees, then they don't interact and you don't draw any link. If they are separated by 120 degrees, then you draw a regular link. If they are separated by 135 degrees, then you draw a double link. If they are separated by 150 degrees, then you draw a triple link.

In your direct sum, the root systems are independent, so the roots in one root system are all at 90 degrees with the roots in the other root system. You don't draw any links in that case.

So then you would just draw three vertices, one for each $A_1$, as @Kosm explained.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.