Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean
$\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and $V(\lambda)$ is the highest weight representation of weight $\lambda$.
One can calculate the weight modules and just take their sum, however I would like something more succinct.
For example, consider the simple case of $sl_2(\mathbb C)$. This Gel'fand model is simply the complex two variable polynomials. One sees this by writing the highest weight representations of $sl_2(\mathbb C)$ as homogeneous polynomials in variables $x,y$ by considering the Leibniz action of $sl_2(\mathbb C)$ on $C\langle x,y \rangle$. By summing these you get the polynomials in two variables.
I find this particularly intuitive. However, in the more general situation of $sl_n$, I don't see how to do this. Note, I am particularly interested in showing they are isomorphic to rings with nicer forms(I don't care to argue about what I mean by nicer, I think we both know).
What I am even more interested in, is this question for quantized universal enveloping algebras, and again a nice simple case would be $U_q(sl_n)$. Again, our simple case, $U_q(sl_2)$ I know and like: the quantum plane, two variable polynomials quotient $xy-qyx$ for parameter $q$.
I know of a paper or two that mention some of these, but none that I have explain how to see this for the general type A case. In particular, papers about the quantum version are especially rare. References are appreciated. I would also appreciate proofs for other specific cases, they might be enlightening.
Note:This coincides with the homogeneous coordinate ring for $sl_n$.
Thanks in advance!
Edit: A large discussion has taken place with Mariano below. He pointed out that my previous language was incorrect, and has helped me identify the correct question that I wished to ask. Hail to the chief! (I hope he doesn't mind I call him chief. :/)
For a semisimple Lie algebra, the representation ring is a polynomial ring, and can be described quite concretely as the invariant ring $\mathbb Z[\Lambda]^W$ of the group algebra $\mathbb Z[\Lambda]$ of the weight lattice $\Lambda$ under the natural action of the Weyl group $W$. In the quantum case with $q$ not a root of unity, the ring has a similar description, as the deformation is not strong enough to mess much with it, in a sense; if $q$ is a root of unity, things are considerably more complicated.
The classical case is treated in pretty much any good text on representation theory of semisimple algebras, in one form or another. For example, the ever great Representation Theory by Fulton and Harris. The quantum case is treated in the corresponding quantum books :)