Title: Half-liberating the unitary groups
Abstract: Uniformly in each dimension, the unitary group has at least three families of operator-algebraic compact matrix quantum supergroups, all of different degrees of being "quantum". The most classical of the three, along with the associated non-commutative complex geometries, was comprehensively studied by Banica and Bichon with the help of a clever use of Gelfand duality. However, this approach no longer works in the other two cases and barely anything was known about them. After providing the relevant definitions and facts about quantum groups, I report on how to use Tannaka-Krein duality in order to exhibit the two remaining quantum supergroups of the unitary group as co-full quotients of infinite wreath graph products of free quantum or classical groups. That settles in particular the question of co-amenability in a prominent special case and suggests a potential strategy for computing the L² Betti numbers of the discrete duals of those quantum groups.