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.

smallquantum group at a root of unity then there really is an R-matrix without completing. – Noah Snyder Nov 19 '10 at 22:48