Recently Mathematicians in harmonic analysis become more and more interested in Locally compact quantum groups and try to transfer concepts from abstract harmonic analysis to the setting of locally compact quantum groups. I know that the first goal of defining the category of LCQGs was extending the Pontryagin duality for locally compact abelian groups. In fact this leads me to study LCQGs and do research in this area. But after a while I wonder whether there exists any application for locally compact quantum groups in the real world for example physics.
Locally compact quantum groups can be used to construct quantum channels, see the recent preprint http://arxiv.org/abs/1210.2738 by Jason Crann and Matthias Neufang.