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I am trying to understand what is a difference between a quantum double and a quantum group? I thought these two were the same and now I have trouble figuring out how are they related?

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    $\begingroup$ Please include definitions/references. The term quantum group is ambiguous. I know about 20 different uses of this terminology. $\endgroup$
    – user160032
    May 20, 2021 at 23:06
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    $\begingroup$ @MathQED: while I agree that in general it is good to include that kind of context, this particular question is essentially asking for definitions and references. $\endgroup$ May 20, 2021 at 23:11
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    $\begingroup$ @MathQED While there are indeed various different meanings of 'quantum group' -- eg Drinfeld--Jimbo quantised universal enveloping (+ its Yangian limit), Felder elliptic, Woronowicz compact -- and I am willing to believe that you know a handful of them, 20 seems a lot. Please enlighten me $\endgroup$ May 21, 2021 at 3:37
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    $\begingroup$ @JulesLamers 20 is clearly just hyperbole to emphasize the point that the term is ambiguous even to what one might call an "expert on quantum groups". I've seen the terminology pop up just about any time someone constructs a generalization/substitute of a group/Hopf algebra that's somehow vaguely "quantum". To the extent that one might think that any quasi-Hopf algebra and/or its representation category is a "quantum group", or maybe only if it's quasitriangular. Here the "vaguely quantum" bit is that the representation category can now have a non-trivial associator. $\endgroup$ May 21, 2021 at 7:19
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    $\begingroup$ @zibadawatimmy Thanks. While I don't disagree, I don't know if the hyperbole is all that useful and necessarily clear to the OP given the level of the question -- whence my comment $\endgroup$ May 21, 2021 at 7:24

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A "quantum group" is a somewhat vague term. I can talk a bit about one class of examples, but there are others. Some standard references are Quantum Groups by Kassel and A Guide to Quantum Groups by Chari and Pressi.

One set of examples that are basically always included are certain Hopf algebras (called Drinfeld-Jimbo quantum groups) $\mathcal{U}_q(\mathfrak{g})$ for $\mathfrak g$ a simple Lie algebra. These are algebras over rational functions in a formal variable $q$. You can think of $\mathcal{U}_q(\mathfrak{g})$ as a $q$-analogue of the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$, and setting $q = 1$ recovers $\mathcal{U}(\mathfrak{g})$. Sometimes we instead think of them as algebras over a ring of formal power series in $\hbar$, with $q = e^\hbar$.

One important property of quantum groups is that they are quasitriangular, which gives a braiding on their module categories. (Actually this isn't quite true for $\mathcal U_q(\mathfrak g)$ because of some stuff with formal power series.) This leads to the Reshetikhin-Turaev construction of link and 3-manifold invariants, among many other things.

The quantum double or Drinfeld double is a way of constructing quasitriangular Hopf algebras from Hopf algebras. In particular, if $\mathfrak b$ is the Lie algebra of an upper Borel subalgebra of $\mathfrak g$, it's easy to define a Hopf algebra $\mathcal U_q(\mathfrak b)$. The quantum double of $\mathcal U_q(\mathfrak b)$ (which is a sort of semidirect product of $\mathcal U_q(\mathfrak b)$ and its dual) is essentially $\mathcal U_q(\mathfrak g)$. One reason to define this way is that the formula for the universal $R$-matrix (the thing giving the quasitriangular structure) drops out automatically; it's very much not obvious how to define it without the quantum double construction.

Some of your confusion might be that the Drinfeld-Jimbo quantum groups are constructed as quantum doubles, but sometimes the generators and relations and the $R$-matrix are given without mentioning this fact.

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  • $\begingroup$ Do you mind expanding your comment that " The quantum double of $\mathcal U_q(\mathfrak b)$ is essentially $\mathcal U_q(\mathfrak g)$"? Or maybe you have a reference about this. $\endgroup$
    – Vik S.
    Jun 9, 2021 at 10:14
  • $\begingroup$ I was referring to the fact that if you take the double of the upper Borel $\mathfrak b$ of $\mathfrak{sl}_2$, you actually get a quantum group $\mathcal U_q(\mathfrak{gl}_2)$. To recover $\mathcal U_q(\mathfrak{sl}_2)$ you need to identify two generators, which is directly analogus to requiring trace $0$. $\endgroup$ Jun 9, 2021 at 12:33
  • $\begingroup$ There are more details in Kassel's Quantum Groups, Chapters IX and XVII. I think this might be covered in some other references as well. $\endgroup$ Jun 9, 2021 at 12:34
  • $\begingroup$ You might be interested in the answers to mathoverflow.net/questions/20683/…: there are more algebraically sophisticated ways of understanding this. $\endgroup$ Jun 9, 2021 at 12:36

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