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By this I mean the specialisation of the quantum group Uq(g) with q a root of unity, and the 'correct' meaning of 'correct' (enclosed in quotations since there isn't necessarily a correct answer) is likely to mean that its category of representations is the 'correct' one.

My suspicion is that I want to take an integral form of Uq(g) defined over ℤ[q±1/2] and base change to the appropriate ring of integers in a cyclotomic field. Having heard of the 'small quantum group' and Lusztigs algebra U dot (notation in his quantum groups book), I suspect the existence of multiple approaches, which diverge at least when an integral form is desired, and hence turn to mathoverflow for clarification and enlightenment.

Afficionados of this type of question can consider it as a continuation in a sequence initiated by this question on the universal enveloping algebra in positive characteristic.

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There are (at least) five interesting versions of the quantum group at a root of unity.

The Kac-De Concini form: This is what you get if you just take the obvious integral form and specialize q to a root of unity (you may want to clear the denominators first, but that only affects a few small roots of unity). This is best thought of as a quantized version of jets of functions on the Poisson dual group. It's most important characteristic is that it has a very large central Hopf subalgebre (generated by the lth powers of the standard generators). In particular, its representation theory is sits over Spec of the large center, which is necessarily a group and turns out to be the Poisson dual group. It also has a small quotient Hopf algebra when you kill the large center.

The main sources for the structure of the finite dimensional representations are papers by subsets of Kac-DeConcini-Procesi (the structure of the representations depends on the symplectic leaf in G*, in particular there are "generic" ones coming from the big cell) as well as some more recent work by Kremnitzer (proving some stronger results about the dimensions of the non-generic representations) and by DeConcini-Procesi-Reshetikhin-Rosso (giving the tensor product rules for generic reps). The main application that I know of this integral form is to invariants of knots together with a hyperbolic structure on the compliment and to invariants of hyperbolic 3-manifolds due to Kashaev, Baseilhac-Bennedetti, and Kashaev-Reshetikhin. The hope is that these invariants will shed some light on the volume conjecture.

The Lusztig form: Here you start with the integral form that has divided powers. Structurally this has a small subalgebra generated by the usual generators (E_i, F_i, K_i) since E^l = 0. The quotient by this subalgebra gives the usual universal enveloping algebra via something called the quantum Frobenius map. The main representation that people look at are the "tilting modules." Tilting modules have a technical description, but the important point is that the indecomposable tilting modules are exactly the summands of the tensor products of the fundamental representations. Indecomposable tilting modules are indexed by weights in the Weyl chamber. The "linkage principle" tells you that inside the decomposition series of a given indecomposable tilting module you only need to look at the Weyl modules with highest weights given by smaller elements in a certain affine Weyl group orbit.

It is the Lusztig integral form (not specialized) that is important for categorification. The Lusztig form at a root of unity is important for relationships between quantum groups and representations of algebraic groups and for relationships to affine lie algebras. The main sources are Lusztig and HH Andersen (and his colaborators). I'm also fond of a paper of Sawin's that does a very nice job cleaning up the literature.

The Lusztig integral form is also the natural one from a quantum topology point of view. For example, if you start with the Temperley-Lieb algebra (or equivalently, tangles modulo the Kauffman bracket relations) and specialize q to a root of unity what you end up with is the planar algebra for the tilting modules for the Lusztig form at that root of unity.

The small quantum group:

This is a finite dimensional Hopf algebra, it appears as a quotient of the K-DC form (quotienting by the large central subalgebra) and as a subalgebra of the Lusztig form (generated by the standard generators). I gather that the representation theory is not very well understood. But there has been some work recently by Roman Bezrukavnikov and others. I also wrote a blog post on what the representation theory looks like here for one of the smallest examples.

The semisimplified category:

Unlike the other examples, this is not the category of representations of a Hopf algebra! (Although like all fusion categories it is the category of representations of a weak Hopf algebra.) You start with either the category of tilting modules for the Lusztig form or the category of finite dimensional representations of the small quantum group and then you "semisimplify" by killing all "negligible morphisms." A morphism is negligible if it gives you 0 no matter how you "close it off." Alternately the negligible morphisms are the kernel of a certain inner product on the Hom spaces. The resulting category is semisimple, its representation theory is a "truncated" version of the usual representation theory. In particular the only surviving representations are those in the "Weyl alcove" which is like the Weyl chamber except its been cut off by a line perpendicular to l times a certain fundamental weight (see Sawin's paper for the correct line which depends subtly on the kind of root of unity).

This example is the main source of modular categories and of interesting fusion categories. Its main application is the 3-manifold invariants of Reshetikhin-Turaev (where this quotient first appears, I think) and Turaev-Viro. For those invariants its very important that your braided tensor category only have finitely many different simple objects.

The half-divided powers integral form:

This appears in the work of Habiro on universal versions of the Reshetikhin-Turaev invariants and on integrality results concerning these invariants. This integral form looks like the Lusztig form on the upper Borel and like the K-DC form on the lower Borel. The key advantage is that in the construction of the R-matrix via the Drinfeld double you should be looking at something like U_q(B+) \otimes U_q(B+)* and it turns out that the dual of the Borel without divided powers is the Borel with divided powers and vice-versa. There's been very little work done on this case beyond the work of Habiro.

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wow. good answer! –  Chris Schommer-Pries Dec 29 '09 at 17:18
Amazingly instructive! I would vote it up ten times if I could. –  André Henriques Sep 9 at 17:08

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