# The Major Families of Quantum Groups

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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The Drinfeld-Jimbo algebras are not quasi-triangular. You need to look at the h-adic versions (see Kassel). –  Jonas Hartwig Nov 19 '10 at 19:06
Even for $q$ not a root of unity?! –  Dyke Acland Nov 19 '10 at 19:20
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). –  Jonas Hartwig Nov 19 '10 at 19:46
One caveat, if you look at the small quantum group at a root of unity then there really is an R-matrix without completing. –  Noah Snyder Nov 19 '10 at 22:48
You might want to have a look at this old answer and some of the links it contains: mathoverflow.net/questions/10036/… –  Noah Snyder Dec 1 '10 at 15:45