# What are $q$-deformations?

This question has already appeared in a lot of different ways and here is another one.

First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. This thing is often called a $q$-deformation of $\mathfrak{sl}_2$. The first baby question is whether anyone can give a more general definition of what a $q$-deformation is without generators and relations.

Immediately I want to give two remarks:

• People wrongly call the quantized universal enveloping algebras $U_q(\mathfrak{g})$ where $\mathfrak{g}$ is a finite-dimensional complex semi-simple Lie algebra, Drinfeld-Jimbo algebras. Why do I say that this is wrong? Well, the algebras $U_h(\mathfrak{g})$ defined in a similar fashion are the real Drinfeld-Jimbo algebras. In chapter XVIII of 'Quantum Groups' by Kassel, he defines $h$-deformations. Moreover, there are rigidity-results for these algebras. One can truly say that a Drinfeld-Jimbo algebra is the unique quantization of the underlying Lie algebra in very precise sense (again see Kassel's book). In fact, the Drinfeld-Jimbo algebras have a nice categorical interpretation (see corollary XIX.4.3 of his book). Many people often claim that these rigidity results hold for the $q$-deformations as well. I have yet to see a proof of such claims! Moreover, I believe that people often claim this because they refer to both $q$ and $h$-deformations as Drinfeld-Jimbo algebras. This explains my wrongly.
• Assuming that these results are easily copied to $q$-setting is not a good hope. The $h$-rigidty results depend on the fact that $h$-deformations $U_h(\mathfrak{g})$ have a quasi-triangular structure. This is simply not true for $U_q(\mathfrak{g})$. (Categorically this correspond to the representation category of $U_h(\mathfrak{g})$ having a braiding whereas $U_g(\mathfrak{g})$-Mod is not braided).

With these considerations in mind, any reasonable definition of a $q$-deformation must be seriously different than the one with generators and relations. Indeed, one can only talk about rigidity meaningfully if $q$-deformations aren't intrinsically uniquely defined (specifying generators and relations tells you what the object is, uniquely).

Lastly, given a nice Lie algebra $\mathfrak{g}$. One can look at the finite-dimensional representations $U(\mathfrak{g})$-Mod. This category is a monoidal category. One should be able to 'monoidally deform' this category in essentially one way. Applying reconstruction theorems to the deformed category should give you a quantization of $\mathfrak{g}$.

What cohomology is used for this monoidal deformation? What is the quantization you recover? Is it an $h$-deformation? Or is this what $q$-deformations should be?

After some looking around, I think the so called Davydov-Yetter cohomology might be an interesting cohomology to look at. I'm having troubles understanding this cohomology though. (Etingof's book on tensor categories explains what it is, but not everything is defined). A good reference on this subject would be appreciated.

• Did you try the books by Chari & Pressley or Fuchs? I don't have either at hand, so I can't check now (but I seem to recall they discuss both cases, so I'd hope they'd answer this too). – Jules Lamers Feb 15 '18 at 19:09
• @JulesLamers: I just skimmed the book Chari & Pressley. In chapter 6 the $h$-deformations (QUE algebras) are introduced much like in Kassel and rigidity results are proven. Chapter 9 introduces $q$-deformations as specializations of QUE algebras, thus again these are introduced by generators and relations. The representation theory of both kind of algebras is discussed, but I couldn't find a general definition of $q$-deformations. So certainly no rigidity results either. – Mathematician 42 Feb 15 '18 at 20:25
• Thanks for the references. I'll try the book of Fuchs when I find it. Perhaps I missed something in Chari & Pressley. If you read the book you might be better qualified to find an answer. – Mathematician 42 Feb 15 '18 at 20:31
• Okay. I have only ever seen presentations via generators and relations. Would be nice if there's an actual definition of a q-deformation, though – Jules Lamers Feb 16 '18 at 16:55
• @JulesLamers: See for example theorem 3.16 in Schiffmann's notes (read the text above the theorem to understand the notation). Theorem 3.16 recovers the positive part of certain quantum groups as Hall algebras of nice categories. One can introduce the Drinfeld double of a Hall algebra to recover the entire quantum group. – Mathematician 42 Feb 17 '18 at 22:17