Cartan-Weyl basis and irreducible representations

Question

I am just starting to look at this concept in particle physics, and wondering if someone could clear something up for me. It concerns irreducible representations and the Cartan-Weyl basis. (I'm sorry if it sounds really ignorant but I'm stuck!)

In physics we almost always work with unitary representations, and since these can be expressed as the direct sum of irreducible representations we basically always work with irreducible representations of our algebras.

Now, I understand that any simple Lie algebra may be put into its Cartan-Weyl basis. But what I am wondering is this: suppose I make an irreducible representation of my algebra. Can I ask: is the Cartan-Weyl basis designed to only be used with irreducible representations? If not, if I start off with an irreducible representation and then put it into Cartan-Weyl form, can I guarantee that it will still constitute an irreducible representation?

Any help would be very happily received.

Answer 1

Your question is not entirely clear and I am interpreting it as asking for bases of irreducible representations.

There are Gelfand-Tsetlin bases. These were initially bases of irreducible representations of GL(n) and SL(n) and then the construction was extended to classical groups.

There are ad hoc bases for certain small representations. For example, quasiminiscule representations have one dimensional weight spaces.

The only general construction I know that works for all irreducible representations and all types is the theory of canonical bases due to Kashiwara and to Lusztig. I suspect you will find this theory intimidating and in any case it is not suited for calculations.