Struggling to understand Lie Algebra: Constructing Weight Diagrams

[semisimple]

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 textbooks are saying, such as killing forms, dual spaces, and any of the tensor things.
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?

 

[matt grime]

It is perfectly possible to understand, and counts as one of the nicest parts of mathematics in my opinion. But, as you say, you are out of your depth, and you really ought to start from the beginning. If you don't know what the dual space of a vector space is then there is no real point in trying to understand the theory of semisimple Lie algebras.

 

[Cexy]

I might be able to be some help, as I'm from a similar background - I'm a physicist taking a course in representation theory of Lie algebras for my exams. Our teacher took a fairly 'applied' approach to the course. Although it was obviously still a course in pure mathematics, with all of the rigour that implies, she also devoted a lot of time to understanding the representations of sl_2 and sl_3 and the corresponding weight diagrams in order to get a 'feel' for doing the work. I can recommend the book 'Representation Theory' by Fulton and Harris which takes a fairly heuristic approach.

As matt says though, there's absolutely no point trying to learn the course until you're well versed in linear algebra - you have to know about dual spaces and the tensor product at least in order to get anywhere at all, and being comfortable with ideals is also important.

 

[semisimple]

Thanks a lot for your replies. I wish I had more time to try to understand, but it is a five week lie algebra course that finishes in a week. I will get the book you recomended out tomorrow and be staying up every night. Honours is hard, especially when you are struggling with coursework. I think when it's all over and I have more time, I'll start from the very beginning like you say, because it's an area I would very much like to understand. But right now I don't have time to start from the beginning. I kind of understand dual space; an element in it turns a line into a dot, or a dot into a line, just like the 'a' in the dot product 'a.b' and the ket in the bra-ket notation. Right?

 

[matt grime]

It doesn't turn lines into dots or dots into lines.

If V is a vector space V* is the Hom(V,k) where k is the underlying field of the vector space. This is like the bra-ket notation, and you can (probably should) think of things in terms of dot products: V* is (for finite dimensional vector spaces) a vector space of the same dimension as V, and there is an 'unnatural' identification of V with V*. Given b in V we can define an element of V*, f_b, and f_b is the linear map f_b(a):=a.b. The identification is unnatrual because although it is a vector space isomorphism of finite dimensional vector spaces it so 'contravariant', which means that if you have a M:V->W a linear map there is an induced map on dual spaces but it goes from W* to V*.

Now, as to your question.

I think i see what you're trying to do: construct the weight lattice.

Let's do sl_2. The cartan matrix is just [2], so the root lattice is 1-dimensional. If we start with a highest weight vector of weight n, then the other weights are indeed n-2,n-4,..,-n

sl_3 has a 2x2 cartan matrix with rows (2,-1) and (-1,2), its root lattice looks like isometric graph paper if you have some to hand: tesselate the plane with equilateral triangles.

I think I see what they're getting at.Right, applying the cols of the cartan matrix is like applying the generators of the lie algebra to the weight vector, you just have to draw on the orbit of a point under this action, which is a fancy way of saying plot all the points you can map to under this transformation. It doesn't matter which order you apply the generators, you just keep applying them and drawing in the points you get to.

You know when to stop because you know when powers of the generators are zero. For instance in sl_2 if you have a highest weight representation of weight T, then there are weight vectors of weight T-2n for all integer n by applying one of the generators to it, and this must terminate: look at the defining relations of the generators, you know [ef], [eh]. [fh] for generators e,f,h, h being the 'cartan' element, ie the element for which these weight vectors are eigenvectors.Anyway, the point is that given some weight vector, you can get the other weight vectors in sub representation it generates by applying the generators of the lie algebra, and the Cartan matrix tells youi exactly what the composition rules are for the generators.