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.

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