Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles

TL;DR

This paper explores the relationship between noncommutative Poisson boundaries and Furstenberg-Hamana boundaries in quantum groups, revealing their equivalence or quotient relations in various cases, including deformations of classical groups.

Contribution

It establishes a connection between Poisson and Furstenberg-Hamana boundaries for quantum groups, extending the concept to representation categories and demonstrating invariance under monoidal equivalence.

Findings

The boundary of D(G_q) for q-deformed compact Lie groups is G_q/T.

The boundaries coincide or relate as quotients in many computed cases.

The construction respects monoidal equivalence and extends to tensor categories.

Abstract

We clarify the relation between noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of quantum groups. Specifically, given a compact quantum group $G$, we show that in many cases where the Poisson boundary of the dual discrete quantum group $\hat G$ has been computed, the underlying topological boundary either coincides with the Furstenberg-Hamana boundary of the Drinfeld double $D(G)$ of $G$ or is a quotient of it. This includes the $q$-deformations of compact Lie groups, free orthogonal and free unitary quantum groups, quantum automorphism groups of finite dimensional C$^*$-algebras. In particular, the boundary of $D(G_q)$ for the $q$-deformation of a compact connected semisimple Lie group $G$ is $G_q/T$ (for $q\ne1$), in agreement with the classical results of Furstenberg and Moore on the Furstenberg boundary of $G_{\mathbb C}$. We show also that the construction of the Furstenberg-Hamana boundary of $D(G)$ respects monoidal equivalence and, in fact, can be carried out entirely at the level of the representation category of $G$. This leads to a notion of the Furstenberg-Hamana boundary of a rigid C$^*$-tensor category.