On Finding Lie Algebra Representations

[antibrane]

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]

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
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-representations/where you can also find as an example the classification of $\mathfrak{sl}(2,\mathbb{C})$ representations and use the corresponding theorems.