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 ${\mathbb{R}\mathbb{P}}^{n-1}$ 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.