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If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups?

Of couse, to start with we have the h-adic completions of the Drinfeld-Jimbo deformations of the enveloping algebras of complex semi-simple Lie algebras $U_q({\mathfrak g})$, and the dually defined quantised coordinate algebra of the compact semi-simple Lie groups. With these two families really being two sides of the same thing.

Apart from this, the only other main family that I know of are Majid's bicrossproduct Hopf algebras, which he says can be thought of as deformations of solvable groups.

Majid has two other constructions called bosination and double-bosination, but I don't know if these are quantum groups in the sense defined above.

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    $\begingroup$ The Drinfeld-Jimbo algebras are not quasi-triangular. You need to look at the h-adic versions (see Kassel). $\endgroup$
    – Jonas Hartwig
    Commented Nov 19, 2010 at 19:06
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    $\begingroup$ Yes, even for $q$ not a root of unity! Of course, the category of (say) finite-dimensional representations of $U_q(\mathfrak{g}$ is braided, and there is an explicit formula for the braiding on two modules. However, that braiding $R_{V,W}$ does not come from a universal R-matrix for $U_q(\mathfrak{g})$, but rather from an element in some completed tensor product. Basically, the expression of the braiding on two modules involves infinite sums (which are such that they make sense on f.d. modules). $\endgroup$
    – Jonas Hartwig
    Commented Nov 19, 2010 at 19:46
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    $\begingroup$ One caveat, if you look at the small quantum group at a root of unity then there really is an R-matrix without completing. $\endgroup$
    – Noah Snyder
    Commented Nov 19, 2010 at 22:48
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    $\begingroup$ You might want to have a look at this old answer and some of the links it contains: mathoverflow.net/questions/10036/… $\endgroup$
    – Noah Snyder
    Commented Dec 1, 2010 at 15:45
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    $\begingroup$ Majid's double bosonization is quasitriangular (at least in a completion if necessary, like for $U_q(\mathfrak{g})$). The same holds for the Drinfeld double which is a special case. Ordinary bosonizations $H\ltimes B$ (or semi-direct products) are quasitriangular if $B$ is quasitriangular as a Hopf algebra object in the braided category of $H$-modules (given $H$ itself quasitriangular). $\endgroup$
    – Zahlendreher
    Commented Jun 27, 2013 at 20:14

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I've been having trouble answering this question because I think your notion of "quantum group" is either too restrictive or too expansive. Hopf algebras suffer from annoying analytic issues as soon as they're infinite dimensional, so you should either be looking at finite dimensional quasitriangular Hopf algebras, or you should pick some particular world of analysis you want to work in (C* quantum groups, h-adic quantum groups, etc.). On the other hand, there's no real reason to restrict your attention to Hopf algebras, lots of things that go under the name "quantum group" (most notably the semisimplified categories at a root of unity which occur in Reshetikhin-Turaev's construction of 3-manifold invariants) are not the category of representations of a Hopf algebra, instead they're a braided tensor category.

Anyway some important constructions that you don't mention include:

  • Drinfel'd twists of already known quasi-triangular Hopf algebras
  • Triangular Hopf algebras (classified by Etingof-Gelaki)
  • Woronowicz-style quantum groups in the C* setting
  • Various flavors of quantum groups at roots of unity
  • Bruguieres-Mueger quotients (often called "modularization" or "deequivariantization) of known braided tensor categories
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  • $\begingroup$ So actually we should focus on the categories related to the quantum group instead of the quantum group itself? $\endgroup$
    – Xiao Xinli
    Commented Dec 1, 2010 at 18:00
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    $\begingroup$ For the most part, yes, it's the category of representations that contains the important information. $\endgroup$
    – Noah Snyder
    Commented Dec 2, 2010 at 0:22

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