Quantum automorphism groups of finite graphs: a survey

Abstract: Based on the theory of $C^*$-algebras, Woronowicz developed an analytic approach to quantum groups in the 1980s. In the 1990s, Sh. Wang defined quantum permutation groups within his framework; these are quantum versions of the well-known symmetric groups. In the 2000s, Banica and Bichon defined the notion of a quantum automorphism group of a finite graph, building on Wang’s quantum permutation groups. Such a quantum group contains the automorphism group of the given graph, but in some cases, it may be strictly larger. So, in a way, we then have more ways of quantum permuting vertices rather than just permuting them - we have more symmetries.

I will briefly introduce to compact matrix quantum groups in the sense of Woronowicz and then survey the current knowledge on quantum automorphism groups of graphs. I will also indicate some open problems in this relatively new field.

combinatoricscomplex variablesfunctional analysisgeneral mathematicsgroup theoryK-theory and homologynumber theoryoperator algebrasprobabilityquantum algebrarings and algebrasrepresentation theory

Audience: advanced learners

GAG seminar

Series comments: Description: Non-specialized research seminar