# Lately in Lie Groups

This material is coming from chapter 21 of Fulton and Harris. Below is mostly a regurgitation with a few clarifying remarks.

Recall axioms for a root system

- spans
- implies ( iff )
- The reflection maps to itself
- is an integer.

##5(Clarifying point) Now . Consider the longest string

by axiom 3 it follows that this string is sent to itself under ; in particular,

so comparing where and get sent, it follows that the highest element gets mapped to the lowest element . From this you get

So . etc. etc.

##6 (clarifying point) are as above and then is a root. If then is a root. If , it should still hold that is a root.

The point is , the subalgebra generated by that’s isomorphic to acts on and it breaks up into a direct sum of irreducible representations that have the property that if and are roots, then everything in between is a root. You might also be able to prove this playing around with reflections .

in any case, it follows that implies is a root and implies is a root.

If and if is a root then so is . This shows are either both roots or both not roots.

##7 if are simple distinct roots, then cannot be roots, otherwise , contradicting simple.

##8 From 6,7 it follows that so the angle between them satisfies , meaning (i.e. not accute)

##9 an excercise in Fulton and Harris (that has hints) shows the simple roots are linearly independent. Basically assume you can write a vector in two different ways using disjoint subsets of the simple roots (i.e. you have a relation), take , show its zero to get a contradiction.

##10 As spans there must be exactly simple roots with consequently form a basis. And its not hard to show any positive root can be written as a nonnegative combination of the simple roots.

The rest of the lecture was about makin Dynkin diagrams and then a big theorem saying all the possible Dynkin diagrams are with 5 exceptions.

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- Published:
- November 8, 2009 / 7:06 pm

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- Lie Groups (course)

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