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. :/)