# Lately in Lie Groups

Rmk: Here is a complex semisimple lie algebra. Making it complex is just so that you always have eigenvalues when you want them.

Most of this material is coming from Appendix C and Appendix D of Fulton and Harris. We proved the absolute Jordan decomposition for a semisimple lie algebra. That is, for with semisimple and you have a faithful representation then then you can use normal linear algebra Jordan decomposition to get etc. In fact come from elements of . These are absolute in the sense that for any other representation, .

We started talking about Cartan Subalgebras. : maximal abelian subalgebras of a semisimple lie algebra that consist only of semisimple elements. In fact we proved that in such a situation, its sufficient that the subalgebra just be maximal with respect to only having semisimple elements, i.e. it will then be abelian. In fact any subalgebra of a semisimple consisting of only semisimple elements will be abelian (rough idea: if this wasn’t the case there would be an element with a nonzero eigenvalue and eigenvector, argue by contradiction …).

(Prop D.3) For a semisiple lie algebra, a regular element is one such that the dimension of

Is minimal. The prop. says that for a minimal element, is a cartan subalgebra.

The proof uses at one point that is then (this is ok), but also claims that acts nilpotently, its not said in the proof, but it seems this claims is appealing to exercise D.8 in the appendix.

So now the story is you have and you decompose it into root spaces

Where the spaces have the following properties (the ‘s are called roots)

- They are 1-dimensional
- they consist of eigenvector for for all
- they are parametrized by functionals , i.e. .
- if is a root so is .
- from 5 it follows that for the killing form and for then .

For the proof in Fulton and Harris of 1, its given that is a root, and then its showed that cannot be a root. So it was a root then could not be root, so cannot be a root.

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- Published:
- October 29, 2009 / 9:39 am

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- Lie Groups (course)

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