# 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 $sl_n$.

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. :/)

## 1 Answer

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 :)

• Actually, the ring presented in Fulton-Harris is not the ring I have asked about. The ring that is presented in F-H is generated over the integers, and by the elementary symmetric polynomials. These coincide with the highest weight vectors for each representation(which is why the scalars are integers). Here I am asking about the representation ring, which is the sum of the representations with dominant highest weight. As I mention above, the example for this ring in sl_2 is C[x,y], but F-H's ring would yield Z[x+y]/(xy-1). I would be extremely interested to see how to move between these rings. – BBischof Mar 14 '11 at 23:33
• @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
• If you have in mind another definition of representation ring, maybe you could define it? Mine is the free abelian group on the f.d. reps. modulo the relation that makes direct sum equal to addition, and product induced by tensor product. – Mariano Suárez-Álvarez Mar 15 '11 at 1:46
• Maybe you are thinking of Gelfand models? – Mariano Suárez-Álvarez Mar 15 '11 at 1:59
• 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