Links to few questions that have already been published on MO:

a) http://mathoverflow.net/questions/86575/why-is-the-quantum-lorentz-group-not-connected
Or: What does it mean for a a (compact) quantum group to be connected? (Ok, the (quantum) Lorentz group is not compact, but I find the question interesting).

b) http://mathoverflow.net/questions/43331/quantum-group-calculations-in-mathematica
Or more generally: What software is available for "quantum group calculations" (in Mathematica, Maple, Sage, GAP, etc.)?

c) http://mathoverflow.net/questions/29810/weyl-character-formula-for-quantum-groups

d) http://mathoverflow.net/questions/17861/bicovariant-calculi-on-the-quantum-unitary-groups

e) http://mathoverflow.net/questions/118124/matrix-model-or-cocycle-twist-construction-for-q-deformations-of-compact-simple-l
Or: Find exemples of compact quantum groups for which are "close" to classical compact groups, e.g. can be constructed via cocycle twists or can be described as matrix-valued functions on classical groups.