# Simple modules for $U_q(\mathfrak{sl}_n)$ at roots of unity

Can anyone point me to a classification/construction of the irreducibles for $U_q(\mathfrak{sl}_n)$, or the associated small quantum groups, when the parameter $q$ is a root of unity and $n>2$? Neither Jantzen or Lusztig's quantum group books seem to help.

Edit: perhaps the best way to clarify what I mean when I refer to the small quantum group' is to give the definition: take the $\mathbb{C}$-algebra (other fields will do) generated by (for $n=3$) $E_1, E_2, F_1, F_2, K_1, K_2$ subject to the following relations.
$$E_1^2 E_2 - [2] E_1E_2E_1 + E_2 E_1^2 =0$$ $$E_2^2 E_1 - [2] E_2E_1E_2 + E_1 E_2^2 =0$$ and the same relations on the $F$s, where $[2]$ is the quantum integer $q+q^{-1}$ $$[E_i, F_j] = \delta_{ij} (K_i-K_i^{-1})/(q-q^{-1})$$ $$K_i E_j K_i^{-1} = q^{a_{ij}} E_j$$ $$K_i F_j K_i^{-1} = q^{-a_{ij}} F_j$$ where $[a_{ij}]$ is the Cartan matrix for $\mathfrak{sl}_3$. $$E_i^N = F_i^N = 0, K_i^N=1$$ and also define $E_{1+2} = qE_2E_1 - E_1E_2$ and $F_{1+2}$ similarly and impose $E_{1+2}^N=F_{1+2}=0$. I'm convinced this last is necessary for the algebra to be finite-dimensional, though I have seen papers omitting it --- the small quantum group above should be a finite dimensional Hopf algebra with dimension $N^8$, with a PBW basis described by Lusztig. $q$ is a primitive $N$th root of unity in the field used.

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It's important to clarify the language, since the literature cited by me and David Hill goes in entirely different directions. The "small quantum group" seems to refer to the construction in Lusztig's important 1990 paper in J. Amer. Math. Soc.. – Jim Humphreys Mar 2 '11 at 13:24
Thanks for adding more detail. By the way, it's usual to require $N \geq 3$ in this case, though Andersen and others have studied the extra complications when $N=2$ (or $N=3$ for type $G_2$). Even though $\mathfrak{sl}_n$ is fairly well-behaved, it's extremely hard in practice to work out explicit results about the (finite dimensional) simple modules for quantum groups even in terms of characters and dimensions when $n>4$'. The modular analogue also needs $N=p$ to be very large compared with the Coxeter number $n$. – Jim Humphreys Mar 2 '11 at 15:45
You're right, I need $N \geq 3$: when $N=2$ the PBW basis breaks down and the dimension formula is wrong. – M T Mar 2 '11 at 15:56

I'm interpreting your quantum group as the quantized enveloping algebra studied by Lusztig and many others, starting with the divided power version of the usual enveloping algebra of a semisimple Lie algebra. There is a different version based on the usual enveloping algebra, studied especially by DeConcini, Kac, Procesi, and their students. The case of $\mathfrak{sl}_n$ is in some ways simpler than the general case, but when the parameter is a root of unity even this quantized enveloping algebra or the finite dimensional version introduced by Lusztig is extremely complicated to study. (What is the center, for example?)

For a broad survey (up to the publication date 2003), take a look at the added Chapter H in Jantzen's AMS book Representations of Algebraic Groups (2nd edition of 1987 book) along with his many references. The original papers of Lusztig from the 1980s onward have been a major source of inspiration, but earlier ones are not on the arXiv.

To be very brief, the parametrization by highest weights of irreducible quantum group representations at a root of unity follows the same outline as in the modular representation theory associated with the same type of algebraic group. This includes an analogue of Steinberg's twisted tensor product theorem, reducing the problem to special weights. The usual Weyl group action on weights here gets enriched to an affine Weyl group action, with associated Kazhdan-Lusztig theory. Natural analogues of the older Kazhdan-Lusztig conjecture eventually got proved, but rather indirectly in terms of affine Kac-Moody theory in characteristic 0. (Lusztig conjectured further that these results would reappear in suitable prime characteristics, which was then largely proved by Andersen, Jantzen, Soergel in a long Asterisque volume.)

Basically the parametrization of representations is reasonable (in terms of weights), while the character formula for irreducibles is in the form of an alternating sum KL Conjecture like the classical one but even more difficult to compute due to the use of affine Weyl groups. On the other hand, construction of these representations (as in the characteristic 0 infinite dimensional theory) is not likely to be a reasonable problem even though the dimensions here are finite.

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Many thanks, I will chase those references. I wonder if it is possible to explicitly write down the irreps on which the $E^N$ and $F^N$ act as zero and the $K^N$ as 1, where $N$ is the order of $q$ (i.e. irreps of the small quantum group'). Surely for $N=3$ at least... Hopefully some of the references will make this clearer to me. – M T Mar 2 '11 at 0:11
The $\mathfrak{sl}_3$ example has been worked out in these settings, say for primes $p \geq 3$ or your parameter $N \geq 3$ in the quantum case, but without explicit constructions of simple modules. The key highest weights lie in two alcoves for the affine Weyl group relative to $N$. Weights in the closure of the lower alcove give characters and dimensions as in Cartan-Weyl theory, but a weight inside the top alcove gives a non-simple "Weyl module". Then subtract the simple module with reflected highest weight in lower alcove as in Lusztig's Conjecture (A-J-S, Asterisque 220). – Jim Humphreys Mar 2 '11 at 13:34

I have tracked down some results on explicit classifications of simple modules for $u_q(\mathfrak{sl}_3)$. The general picture is that the simple modules are bigraded by the root lattice and look like towers of concentric hexagons.

For the benefit of anyone else interested, there is a long series of papers by Dobrev:

Multiplet classification of highest weight modules over quantum universal enveloping algebras: the Uq(SL(3,C)) example in Groups St Andrews 1989 vol 1 LMS LNM #159

Representations of Quantum Groups, Symmetries in Science V (Lochau 1990), 93–135, Plenum Press, NY, 1991.

A chapter from Lecture Notes in Physics, 1990, Volume 370, here

...and many others.

also there is a paper by Abdesselam, Arnaudon, Chakrabarti and a discussion of dimensions by Pereira here

Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.

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It's interesting to see how such things are approached by a mathematical physicist. Maybe it's also worth mentioning other mathematical sources for low rank examples in work of Nanhua Xi (Beijing), such as Maximal and primitive elements in Weyl modules for type $A_2$. J. Algebra 215 (1999), no. 2, 735–756. (One of those commercial journals ...) But type $A_2$` is misleading in some ways due to the nice behavior of primitive elements as well as multiplicity 1 composition factors. Concrete calculations are definitely harder than in classical cases. – Jim Humphreys May 21 '11 at 15:42
many thanks for pointing out Xi's paper, which has the full submodule structure of the Vermas for a certain Frobenius kernel. – M T May 24 '11 at 21:15

Look at this paper of Chari and Pressley.

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I linked the arXiv version instead. – David Hill Mar 1 '11 at 20:56
Another reference: MR1757013 by Cantarini. – David Hill Mar 1 '11 at 21:04
thanks for the links David. – M T Mar 2 '11 at 0:04