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.
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2$\begingroup$ Quantum groups do arise in quantum physics and are useful there. See the answer to this question: mathoverflow.net/questions/16024/… Take a look at the references in the answer - my impression is that physicists work only with hopf algebras and do not look at any topological structures, so you may have to work to see what they're doing from an operator algebra point of view. I was planning on doing this at some point, but it's on a long list of "to do" items $\endgroup$– OllieCommented Oct 17, 2012 at 17:03
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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.