Building positive roots from simple roots

In summary, the conversation discusses the decomposition of a semisimple Lie algebra into root spaces, and the relationship between the roots of individual elements and their commutator. The conversation also mentions an "inductive" method for building positive roots from simple ones, and raises the question of whether this method may miss certain roots if the commutator of simple roots results in a zero root.
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alexgs
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Hi, I've got two related questions.

You can decompose a (semisimple) lie algebra into root spaces, each of which are 1-dimensional. If X has root a and Y has root b then [X,Y] has root a+b. If the root space of a+b is not zero (i.e. there is a root a+b) then is it possible for [X,Y] to still be 0?

I ask because in Georgi on p.107 he has an "inductive" method of building all the positive roots from the simple ones where, if X,Y,... are the simple roots you just form all the brackets of them and check which ones are roots.

For instance, if X and Y are simple you want to see if there are any positive roots a+b. So you form [X,Y] and then check to see if it's a root. But isn't it possible that [X,Y]=0 even if the root space of a+b is not empty? In this case, wouldn't the "algorithm" miss the root a+b?

Thanks in advance.
Alex
 
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1. What are simple roots?

Simple roots are the basic, indivisible components of a complex root. They cannot be simplified any further and are typically denoted by a single letter, such as "a" or "b".

2. How do you build positive roots from simple roots?

To build positive roots from simple roots, you can use operations such as addition, subtraction, multiplication, and division. For example, if you have the simple roots "a" and "b", you can build the positive root "a + b" by adding them together.

3. Why is it important to build positive roots from simple roots?

Building positive roots from simple roots helps to simplify complex expressions and make them easier to work with. It also allows us to solve equations and analyze functions more efficiently.

4. Can you provide an example of building positive roots from simple roots?

Sure, let's say we have the simple roots "x" and "y". We can build the positive root "x + 2y" by adding them together. This would simplify the expression and make it easier to work with.

5. Are there any rules or guidelines for building positive roots from simple roots?

Yes, there are a few rules to keep in mind when building positive roots from simple roots. For addition and multiplication, the order of the simple roots does not matter. However, for subtraction and division, the order does matter. Additionally, you can only combine simple roots that have the same index (exponent) and radicand (number under the radical sign).

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