# Probabilistic boundaries of finite extensions of quantum groups

## TL;DR

This paper investigates the boundaries of quantum groups and demonstrates that, under certain conditions, the harmonic functions and boundaries of a quantum group relate closely to those of its quotient, with applications to quantum group duals.

## Contribution

It establishes that harmonic functions on quantum groups with finite normal subgroups are invariant and that their boundaries coincide with those of the quotient, extending classical boundary results to quantum settings.

## Key findings

- Harmonic functions are G-invariant on quantum groups with finite normal subgroups.
- Poisson and Martin boundaries of quantum groups match those of their quotients under certain conditions.
- Boundaries of duals of group-theoretical easy quantum groups are classical.

## Abstract

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.