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The rank $n$ swapping algebra is the Poisson algebra defined on the ordered pairs of points on a circle using the linking numbers, where a subspace of $(mathbb{K}^n times mathbb{K}^{n*})^r/operatorname{GL}(n,mathbb{K})$ is its geometric mode. In this paper, we find an injective Poisson homomorphism from the Poisson algebra on Grassmannian $G_{n,r}$ arising from boundary measurement map to the rank $n$ swapping fraction algebra.
F. Labourie [arXiv:1212.5015] characterized the Hitchin components for $operatorname{PSL}(n, mathbb{R})$ for any $n>1$ by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introd
The {em rank $n$ swapping algebra} is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of $(mathbb{K}^n times mathbb{K}^{n*})^r/operatorname{GL}(
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 algebr
We show that the $imath$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $imath$Hall algebra to the ring of s
We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H otimes gl(n,C)$. Then we