On Finding Lie Algebra Representations

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What I am trying to do is start with a Dynkin diagram for a semi-simple Lie algebra, and construct the generators of the algebra in matrix form. To do this with su(3) I found the root vectors and wrote out the commutation relations in the Cartan-Weyl basis. This gave me the structure constants of the algebra and then I was able to write down the matrices corresponding to the regular (adjoint) representation.

What I am confused about (assuming the above is a correct method) is how to find different matrix representations than this. The representation I am familiar with for su(3) consists of the "Gell-Man matrices" which are 3 dimensional and obviously my matrices from my construction are 8 dimensional.

How would I find different representations starting from the commutation relations? I know how to find them starting with the Lie Group. Let me know if I need to clarify something--thanks a lot.
 

fresh_42

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With the commutator relations you have all which is necessary. Any finite dimensional representation is a Lie algebra homomorphism ##\varphi\, : \,\mathfrak{g} \longrightarrow \mathfrak{gl}(V)##. This is a system of quadratic equations and you are looking for the solutions. If ##A_X## is the matrix of ##\varphi(X)## we have to solve ##[A_X,A_Y]=A_{[X,Y]}## for all ##X,Y##. However, this will be a lot of manual work to do.

It is easier to use a book about the representations of Lie Algebras, see e.g. the source list in
where you can also find as an example the classification of ##\mathfrak{sl}(2,\mathbb{C})## representations and use the corresponding theorems.
 

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