# Indecomposable, non-simple, modules of quantum groups at roots of unity

Let us consider the quantum group $$U_q(\mathfrak{sl}_2)$$ (as defined in Kassel's book on quantum groups), for $$q$$ being a root of unity of order $$d$$ (i.e., $$d$$ is the smallest positive integer for which $$q^d=1$$). If $$n$$ is the dimension of an irreducible, finite-dimensional representation of $$U_q(\mathfrak{sl}_2)$$ (over a complex vector space), then it is known that $$n$$ is bounded above by $$e=\begin{cases} d, & \text{d: odd} \\ d/2, & \text{d: even.} \end{cases}$$ As far as I know, there are indecomposable, non-simple modules of dimension higher than $$e$$. I have made some small search on the structure of such modules, but I have not found anything substantial apart from Chari and Premet - Indecomposable restricted representations of quantum $$sl_2$$ (pdf abstract MSN), which however refers to the restricted case. So my questions are:

1. Is there some reference on the structure of indecomposable, non-simple modules of quantum groups at roots of unity?
2. Are there infinite dimensional, indecomposable, non-irreducibles?
3. How can the limits of such representations (either fin or inf dimensional) at $$q\to 1$$, be computed?

I would be interested either on references or on some short—if possible-—description of such modules, mainly for the case of $$U_q(\mathfrak{sl}_2)$$ and more generally for $$U_q(\mathfrak{g})$$, where $$\mathfrak{g}$$ is a finite-dimensional, simple, complex Lie algebra.

Related:

• I suppose that the content behind the link link https://www-ems-ph-org.ezproxy.tcu.edu/journals/show_abstract.php?issn=0034-5318&vol=30&iss=2&rank=11 is only accessible to users from TCU. – Martin Sleziak Aug 17 '19 at 6:55

In the case of the rank 1 simple Lie algebra, your references give a good account of what is known. But in general, it's wise to keep in mind that many of the indecomposable $$U_q(\mathfrak{g})$$-modules resemble those of the universal enveloping algebra of $$\mathfrak{g}$$ in prime characteristic (as seen in many papers by Lusztig, Andersen, et al). Verma module analogues give for example a positive answer to your Question 2. Analogues of tilting modules and the like show the variety of examples available, and there are apparently many quotients, etc. So a listing of cases may be impossible in general.

You might however try the papers by Maxim Vybornov.

• Prof. Humphreys, thank you for your feedback. – Konstantinos Kanakoglou Apr 26 '19 at 2:35

I apologize for self-citation, however in an old paper of mine, together with R. Giachetti The two-dimensional Euclidean quantum algebra at roots of unity, in the process of describing decomposition of tensor products of irreps at roots of unity we listed some explicit indecomposable modules for $$E_q(2)$$.

Let me mention that this is a way in which irreps at roots of unity behave like infinite-dimensional ones: in that he tensor products of irreps are not completely reducible any more (and this happens each time the dimension exceeds the roots of unity degree, which then, in a way, plays the role of $$\infty$$).

I think you may find something of this kind also in papers referring to tensor product decompositions for $$SL_q(2)$$ at roots of unity.

• @Nicola Ciccoli: It's not a matter if self-citation, but rather a question as to which version of "quantum group" is being asked about by Konstantinos. Unfortunately, the relationship between the versions has apparently not been made explicit anywhere. You might look at my answer to a much earlier question mathoverflow.net/questions/57027/… – Jim Humphreys Apr 28 '19 at 20:09
• @Jim Humphreys, my work was in the same setting of De Concini - Kac-Procesi approach to qg's at roots of unity. I agree that the relation between various approaches is at times subtle and overlooked in a significant part of the published literature. – Nicola Ciccoli Apr 28 '19 at 20:14