From Groups to Quantum Groups

In this talk, the theory of operator algebraic quantum groups will be outlined, which are generalisations of groups and are key to extensions of Pontryagin Duality, and for encoding quantum symmetries of classical and quantum spaces, more recently with deep applications in quantum information theory. A striking example is Wang's Quantum Permutation groups, while classical permutation groups, which encode Symmetries of finite sets are finite, and the Permutation quantum groups are infinite-dimensional objects. This talk will be an introduction to the theory of (compact) quantum groups with some examples and properties.