Quantum double vs Quantum group

Question

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?

Answer 1

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.