Links to few questions that have already been published on MO:
a) 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) Quantum Group Calculations in Mathematica Or more generally: What software is available for "quantum group calculations" (in Mathematica, Maple, Sage, GAP, etc.)?
c) Weyl Character Formula for Quantum Groups
d) http://mathoverflow.net/questions/17861/bicovariant-calculi-on-the-quantum-unitary-groups
e) Matrix model or cocycle twist construction for q-deformations of compact simple Lie groups in $q=-1$? 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.