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We induce a Poisson algebra $\{\cdot,\cdot\}_{\mathcal{C}_{n,N}}$ on theconfiguration space $\mathcal{C}_{n,N}$ of $N$ twisted polygons in$\mathbb{RP}^{n-1}$ from the swapping algebra \cite{L12}, which is foundcoincide with Faddeev-Takhtajan-Volkov algebra for $n=2$. There is anotherPoisson algebra $\{\cdot,\cdot\}_{S2}$ on $\mathcal{C}_{2,N}$ induced from thefirst Adler-Gelfand-Dickey Poissson algebra by Miura transformation. Byobserving that these two Poisson algebras are asymptotically related to thedual to the Virasoro algebra, finally, we prove that$\{\cdot,\cdot\}_{\mathcal{C}_{2,N}}$ and $\{\cdot,\cdot\}_{S2}$ are Schoutencommute.

Swapping algebra, Virasoro algebra and discrete integrable system