i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the von Neumann algebraic quantum group $SU_q(2)$, which will be denotes by $\mathscr{L}^{\infty}(SU_q(2))$ is $B(\ell^2(\mathbb{N}))\overline{\otimes}\mathscr{L}(\mathbb{Z})$ (where the tensor product denotes the von Neumann algebraic tensor product). Can one generalize this to $SU_q(n)$. Can one find an explicit expression for $\mathscr{L}^{\infty}(SU_q(n))$ as above?

Thank you very much