Hi, I have spent all weekend reading Textbooks, where I concentrated on Cahn, trying to understand what is going on in Lie Algebra lecture notes. I am having a lot of trouble because I have no background in maths other than applied maths, and lie algebras is so different to applied maths and I don't understand half of what the text books are saying, such as killing forms, dual spaces, and any of the tensor things.(adsbygoogle = window.adsbygoogle || []).push({});

What I am trying to understand is how to construct weight diagrams. The one's my teacher uses start with column vector (1 0), say, and then he applies F1 or F2 to them where F1 subtracts the first column of the Cartan Matrix from the highest weight and F2 subtracts the second column of the Cartan matrix of the highest weight (for B2 matrix). This is as opposed to the geometric pictures that are in most books and the Young's diagrams, neither of which I understand (but have tried to).

What I am stuck on is how do you know whether you can apply F1 or F2 to a given vector? I can't seem to find any pattern or reasoning or rules, F1 and F2 seem to be applied arbitrarily. For example, for the highest weight column vector (0 2) or (0 1) (transpose), why can't I apply F1? I also don't understand how to know the multiplicity without knowing Freudenthal's formula, which the teacher hasn't taught us but is in the books I've got from library as a way to determine the multiplicity (and I don't understand because it is built on countless other things I don't understand).

Also, I don't understand how to know where the string terminates, because in an example of (2 0) (for the B2 Cartan matrix) (dimension 10), F1 is applied to (0 -1) to get (-2 0) at which point it terminates, and I thought the lowest weight vector was the one with no positive entries, so it should terminate at (0 -1).

Yes, I am out of my depth. Is it possible to ever understand all of this, or is it only for genius's?

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# Weight diagrams

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