Swapping algebra, Virasoro algebra and discrete integrable system

We induce a Poisson algebra ${cdot,cdot}_{mathcal{C}_{n,N}}$ on the configuration space $mathcal{C}_{n,N}$ of $N$ twisted polygons in $mathbb{RP}^{n-1}$ from the swapping algebra cite{L12}, which is found coincide with Faddeev-Takhtajan-Volkov algebra for $n=2$. There is another Poisson algebra ${cdot,cdot}_{S2}$ on $mathcal{C}_{2,N}$ induced from the first Adler-Gelfand-Dickey Poissson algebra by Miura transformation. By observing that these two Poisson algebras are asymptotically related to the dual to the Virasoro algebra, finally, we prove that ${cdot,cdot}_{mathcal{C}_{2,N}}$ and ${cdot,cdot}_{S2}$ are Schouten commute.