From Lie algebras to Dynkin diagrams and back again

[tom.stoer]

I am looking for a free online-resource sketching
i) the way from Lie algebras to root systems and classification via Dynkin diagrams and
ii) back to the Lie Algebra via reconstruction based on the information encoded in the Dynkin diagram.

I would prefer a short PDF or web page, not a huge book :-)

My main interest is to understand what goes wrong, which steps fail or have to be generalized when starting ii) with a Dynkin diagram not contained in the A-, B-, C-, D-, E- F, G-series, e.g. E₉ or E₁₀.

 

[fresh_42]

I am looking for a free online-resource sketching
i) the way from Lie algebras to root systems and classification via Dynkin diagrams and
ii) back to the Lie Algebra via reconstruction based on the information encoded in the Dynkin diagram.

I would prefer a short PDF or web page, not a huge book :-)

My main interest is to understand what goes wrong, which steps fail or have to be generalized when starting ii) with a Dynkin diagram not contained in the A-, B-, C-, D-, E- F, G-series, e.g. E₉ or E₁₀.

Would be easier to simply name a book (found on https://www.amazon.com/dp/0387900535/?tag=pfamazon01-20for $22 - used). Unfortunately the previews I found on Springer's or Google.Books websites didn't include the crucial part I was looking for. However, I've found this pdf which looks quite promising (~p. 50ff.)

http://www.mat.univie.ac.at/~cap/files/LieAlgebras.pdf

The short answer is: It is because of geometry. (But I don't remember the details.)

 

[tom.stoer]

Thanks a lot; I'll have a look at that