# Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$

Let $$V_N$$ denote the $$N$$-dimensional representation of the quantum group $$U_q(\mathfrak s\mathfrak l_2)$$. I am told that in the limit $$N\to\infty$$ with $$q=e^{2\pi i/n}$$ and $$N/n\to\alpha\in(0,1)$$, the representation $$V_N$$ of $$U_q(\mathfrak s\mathfrak l_2)$$ "converges" in some precise sense to an infinite-dimensional representation of the complex Lie group $$\operatorname{SL}_2(\mathbb C)$$.

What is the precise formulation of this result, and where is it discussed in the literature? [My searches so far have not found anything substantial.]

• I think arxiv.org/abs/hep-th/0312282v2 contains something like what you want, but for $N=n.$ Unfortunately I wasn't able to find anything more general/precise by looking through papers that reference it. – dhy Oct 1 '18 at 3:03

This is a very interesting question. I have also made some search but i have not found this result explicitly mentioned somewhere in the literature. However, i remember i have heard such a claim in the past. So i will try to record my thoughts on the problem. Here is the way i understand it:

Since you are considering the limits $$N\to \infty$$ and $$N/n\to\alpha\in(0,1)$$, this implies that $$n\to\infty$$ thus $$q\to 1$$. In this limit: $$U_q(\mathfrak{sl}_2)\stackrel{q\to 1}{\longrightarrow} U(\mathfrak{sl}_2)$$ and so you get a representation of the UEA and thus of the lie algebra $$\mathfrak{sl}_2$$. Since the corresponding lie group $$SL(2,\mathbb{C})$$, is simply connected, its category of representations is equivalent to the category of representations of the lie algebra $$\mathfrak{sl}_2$$ (in the sense that there is a bijection -up to isomorphism- between the lie group and the lie algebra representations). Thus, we finally arrive at a representation (up to isomorphism) of the lie group $$SL(2,\mathbb{C})$$. And since $$N\to\infty$$, the limit -if it exists- will be an infinite dim representation.

Let us try to examine the situation a little closer and attempt to establish whether -and when- the limit is well defined and exists :

The "convergence" of the quantum group to the lie algebra: A non-trivial point in the preceding argument is how to obtain a rigorous formulation of the limit:

$$U_q(\mathfrak{sl}_2)\stackrel{q\to 1}{\longrightarrow} U(\mathfrak{sl}_2)$$

This is a subtle point, which -imo- does not have a unique answer; it depends on the description one uses for the quantum group $$U_q(\mathfrak{sl}_2)$$. Generally, we can either view $$U_q(\mathfrak{sl}_2)$$ as a $$\mathbb{C}[[h]]$$-algebra and write $$q=e^{h/2}$$ and $$K=q^H$$ (in which case $$q\to 1\Leftrightarrow h\to 0$$ and we use Del' Hospital for the $$E$$-action limit and power series expansion for the $$K$$-action limit), or we can view $$U_q(\mathfrak{sl}_2)$$ as a suitable quotient of a larger $$\mathbb{C}$$-algebra, which is however well defined at $$q=1$$. Both methods are well-known. See for example:
- How $$U_{q}(\mathfrak{sl}_{2})$$ becomes the universal enveloping algebra $$U(\mathfrak{sl}_{2})$$ of $$\mathfrak{sl}_{2}$$,
- Quantized Enveloping Algebras at $q=1$,
- Exponentiations over the quantum algebra $$U_{q}(\mathfrak{sl}_{2})$$ (see the discussion and the computations of section 10, p. 65)

The "convergence" of the representation: Let us study the limit on the irreducible, f.d. representations of $$U_{q}(\mathfrak{sl}_{2})$$ with $$q=e^{2\pi i/n}$$:
(First recall that if $$q$$ is not a root of unity, then the f.d., irred, are highest weight representations. They are parameterized by $$\varepsilon=\pm 1$$ and the positive integers, i.e. we will denote such $$N$$-dim, irred, modules as $$V_{\varepsilon, N-1}$$. The notion of the $$N\to\infty$$ limit here is well defined in the following sense: The $$V_{1,N-1}$$ modules here have matrix elements which are not generally continuous functions of $$q$$ but they can be handled with methods similar to those mentioned above to show that

$$V_{1,N-1}\stackrel{q\to 1}{\longrightarrow}V_{N-1}$$

where $$V_{N-1}$$ are the usual $$N$$-dim, irred, highest weight reps of $$\mathfrak{sl}_2$$. The $$V_{-1,N-1}$$ modules "vanish" when $$q\to 1$$. One can be more precise at that point but it would be irrelevant to the rest. We can equally well consider that we are taking the $$N\to\infty$$ limit considering the $$V_{1,N-1}$$ family only).
Now, let us come back to the root of unity case: $$q=e^{2\pi i/n}$$. It is well known that, the irreducible modules now have an upper bound in their dimension: $${\small e=\begin{cases} n, & \text{n: odd} \\ n/2, & \text{n: even.} \end{cases}}$$. It is also well known that the $$N$$-dim irred, representations, now fall into two classes: First, representations for which $$N. These are of the form $$V_{\varepsilon,N-1}$$, with $$\varepsilon=\pm 1$$, i.e. they are exactly the same modules as in the non-root of unity case; i.e. the highest weight modules. However, the second class includes irred. reps with dimension $$N=e$$. These look quite different than the highest weight modules; they generally fall into distinct subclasses including cyclic, semi-cyclic, etc modules. So it would be wise to exclude these $$\dim V_{N-1}=N=e$$ modules from the limitting procedure (if we want some reasonable sense of convergence).
Thus, while $$N\to\infty$$, it is important to keep in mind that $$N. This is -imo- the meaning of the $$\alpha$$ parameter: $$N/n\to\alpha\in(0,1)$$, which in some sense expresses the "relative rate of divergence" between the dimension $$N$$ of the rep and the order $$n$$ of the root of unity $$q$$.

Let us now attempt to compute these limits; i will not include details on the handling of the order of the limits $$N/n\to\alpha$$, $$n\to\infty$$, since i admit i have not -yet- been able to obtain a rigorous formulation of the following but here is what i have got:

• First consider the limit $$N/n\to\alpha$$. We get the result:

$$V_{1,N-1}\stackrel{N/n\to \alpha}{\longrightarrow}V\big(q^{n\alpha}\big)=V\big(e^{2\pi i\alpha}\big)$$

where $$V_{1,N-1}$$ is the $$N$$-dimensional, irreducible, $$U_q\big(\mathfrak{sl}(2)\big)$$-module of highest weight $$\lambda=q^{N-1}$$ and the $$V\big(q^{n\alpha}\big)$$ is the $$U_q\big(\mathfrak{sl}(2)\big)$$-Verma module of highest weight $$\lambda=q^{n\alpha}=e^{2\pi i\alpha}$$.
The $$U_q\big(\mathfrak{sl}(2)\big)$$-action for the $$V_{1,N-1}$$ module is given by: $$K\cdot v_p=q^{N-2p-1}v_p, \ \ E\cdot v_p=[N-p]_qv_{p-1}, \ \ F\cdot v_p=[p+1]_qv_{p+1}$$ whereas its $$N/n\to\alpha$$ limit, gives the $$U_q\big(\mathfrak{sl}(2)\big)$$-action for the $$V\big(q^{n\alpha}\big)$$-Verma module: $$K\cdot v_p=q^{n\alpha-2p-1}v_p, \ \ E\cdot v_p=[n\alpha-p]_qv_{p-1}, \ \ F\cdot v_p=[p+1]_qv_{p+1}$$ where $$[x]_q=\frac{q^x-q^{-x}}{q-q^{-1}}$$, as usual.

• The next step will be to compute the Verma module $$V\big(q^{n\alpha}\big)$$ limit at $$q\to 1\Leftrightarrow n\to\infty$$;
Conjecture: This produces a $$U\big(\mathfrak{sl}(2)\big)$$-Verma module $$V({\small 4\pi i \alpha-1})$$:

$$V\big(q^{n\alpha}\big)\stackrel{n\to\infty}{\longrightarrow}V({\small 4\pi i \alpha-1})$$

with $$U\big(\mathfrak{sl}(2)\big)$$-action given by: $$H\cdot v_p=-(4\pi i\alpha-2p-1)v_p, \ \ E\cdot v_p=-(4\pi i\alpha-p)v_{p-1}, \ \ F\cdot v_p=(p+1)v_{p+1}$$ This is an infinite dimensional, Verma module, of highest weight $$4\pi i \alpha-1$$, for the (undeformed) universal enveloping algebra $$U\big(\mathfrak{sl}(2)\big)$$.

In any case, if there is not some silly mistake in my conjecture (the result comes from a mixture of computations and intuition), we finally arrive at an infinite dimensional $$U\big(\mathfrak{sl}(2)\big)$$-Verma module and from there we can get the corresponding infinite dimensional, $$SL(2,\mathbb{C})$$ representation. I will try to come back if -and when- i will be able to obtain a somewhat more rigorous formulation of the last limit.

• A couple of comments: (1) in order for the limit to "converge", it is important to fix the parity of $N$ as it tends to $\infty$. (2) In order for the limit to be a representation as opposed to a representation-up-to-isomorphism one can fix, throughout the limiting process, a trivialization of one of the (one-dimensional) weight spaces -- e.g. the zero-weight space (assuming $N$ has been chosen to always be odd). – André Henriques Apr 22 '19 at 11:13
• 37 edits in as many days seems excessive – Yemon Choi May 24 '19 at 4:38