The notion of monoidal equivalence for compact quantum groups was recently introduced by Bichon, De Rijdt and Vaes. In this paper we prove that there is a natural bijective correspondence between actions of monoidally equivalent quantum groups on unital $C^{*}$-algebras or on von Neumann algebras. This correspondence turns out to be very useful to obtain the behavior of Poisson and Martin boundaries under monoidal equivalence of quantum groups. Finally, we apply these results to identify the Poisson boundary...

Let $U$ be the two-parameter quantized enveloping algebra $U_{r,s}\left({\mathrm{𝔰𝔩}}_2\right)$ and $F\left(U\right)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F\left(U\right)$ in the case when $r{s}^{-1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.