Quantum automorphism groups of connected locally finite graphs and quantizations of finitely generated groups

Abstract: I present a joint work with Lukas Rollier. We construct the quantum automorphism group of any connected locally finite, possibly infinite, graph as a locally compact quantum group that has the classical (locally compact) automorphism group as a closed quantum subgroup. For finite graphs, we get the quantum automorphism group of Banica and Bichon. One of the key tools is the construction of a unitary tensor category associated with any connected locally finite graph. When this graph is the Cayley graph of a finitely generated group, the associated unitary tensor category has a canonical fiber functor. We thus also obtain a quantization procedure for arbitrary finitely generated groups. In the particular example of groups defined by a triangle presentation, this construction gives the property (T) discrete quantum groups from earlier joint work with Valvekens.

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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