QUANTUM SYMMETRY GROUPS AND RELATED TOPICS WINTER SCHOOL ON OPERATOR SPACES, NONCOMMUTATIVE PROBABILITY AND QUANTUM GROUPS MÉTABIEF, DECEMBER 2014

Groups first entered mathematics in their geometric guise, as collections of all symmetries of a given object, be it a finite set, a polygon, a metric space or a differential manifold. Original definitions of quantum groups, also in the analytic context, had rather algebraic character. In these lectures we describe several examples of quantum symmetry groups of a given quantum (or classical) space. The theory is based on the concept of actions of (compact) quantum groups on C∗-algebras and viewing symmetry groups as universal objects acting on a given structure. Initiated by Wang in 1990s, in recent years it has been developing rapidly, exhibiting connections to combinatorics, free probability and noncommutative geometry. In these lectures we will present both older and newer research developments regarding quantum symmetry groups, discussing both the general theory and specific examples. I would like to thank here all my collaborators on articles related to quantum symmetry groups and thus also to these lectures. Particular thanks are due to Teo Banica, but I would also like to mention Jyotishman Bhowmick, Debashish Goswami and Piotr So ltan.

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