# Lately in Lie Groups

So is a complex semisimple lie algebra so can write

For , and , then nilpotent, so is well defined. is the subgroup gen. by these.

Thm D.22 (Fult. and Harr) If are Cartan, then they are conjugate, and .

Notes: The actual statement is that there is such that . But . And using that we see that the action of is just to conjugate by the elements \exp(X_i)$.

Also so for any , it true .

### Weyl Group stuff

(this is coming from around page 201 of Ful and Harr)

When talking about roots , there is the following chain

where is the lattice generated by the roots, and is the lattice generated by functionals that are integer valued on all , where is the unique element of such that .

Without much motivation define

Its straightforward to verify that this doesn’t intersect the line . So now define the involution associated to

The hyperplane and are both preserved by . The Weyl Group associated to is the group generated by . More generally (maybe), one can define

where is an inner product that makes preserve and reflect the line .

On page 492 of Ful and Harr there is a remark saying that all these results about Complex lie algebras can be extended to any char 0 and alg. closed field by some base change argument (what is it?). Also, given this other results then hold for any char 0 field….

A root is simple if it cannot be written as a sum of other roots. Now back to material from appendix D. A cool fact is

Lemma D.27 The Weyl group is generated by where is a simple root.

This result uses lemma D.25, which says for such , maps to itself.

the roots we’re interested in look like with simple and nonnegative. Now but why is it in ? Also, if is simple and positive, then is minus some integer multiple of , so

why is this still positive?

## About this entry

You’re currently reading “Lately in Lie Groups,” an entry on Math Meandering

- Published:
- November 1, 2009 / 3:40 pm

- Category:
- Lie Groups (course)

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