Probabilistic boundaries of finite extensions of quantum groups

Given a discrete quantum group H with a finite normal quantum subgroup G, we show that any positive, possibly unbounded, harmonic function on H with respect to an irreducible invariant random walk is G-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of H coincide with those of H/G. A similar result is also proved in the setting of exact sequences of C∗-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical. © 2017 World Scientific Publishing Compan