# How to compute the Gel'fand Models for a (quantum) Lie Algebra

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 $sl_n$.

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.
• @BBischof: I had in mind Theorem 23.24 in F-H. The action of the Weyl group I mention above is not by permutation of the variables, so the resulting invariant ring is not generated by elementary symmetric functions (in fact, its elements are Laurent polynomials; in the case of $sl_2$, it is precisely the set of polynomials invariant under $t\mapsto t^{-1}$, why you can easily show to be identifiable with characters of modules) – Mariano Suárez-Álvarez Mar 15 '11 at 1:44
• Consider the action of $sl_2$ on the standard basis of $C<x,y>$ by the liebniz action. This makes a highest weight repesentation with weight given by power of $x$. The action moves the power to $y$. Thus, taking the ring generated by these weight spaces is $C[x,y]$. In this sense, $R(sl_2)=\oplus_{\lambda\in P^+}V(\lambda)$ for $P^+$ the dominant weights and $V(\lambda)$ the corresponding weight representation. This is what I mean. I hope is is clear. Also, I am typing on the iPad, sorry if the Tex sucks. – BBischof Mar 15 '11 at 3:00