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Register a new userRank $n$ swapping algebra for Grassmannian
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Zhe Sun
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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.
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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 introduce the rank $n$ swapping algebra, which is the quotient of the swapping algebra by the $(n+1)times(n+1)$ determinant relations. The main results are the well-definedness of the rank $n$ swapping algebra and the cross-ratio in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank $n$ swapping algebra generated by these cross-ratios to characterize the $operatorname{PSL}(n, mathbb{R})$ Hitchin component for a fixed $n>1$. We also show the relation between the rank $2$ swapping algebra and the cluster $mathcal{X}_{operatorname{PGL}(2,mathbb{R}),D_k}$-space.
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 $(mathbb{K}^n times mathbb{K}^{n*})^r/operatorname{GL}(n,mathbb{K})$. For any ideal triangulation of $D_k$---a disk with $k$ points on its boundary, using determinants, we find an injective Poisson algebra homomorphism from the fraction algebra generated by the Fock--Goncharov coordinates for $mathcal{X}_{operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifraction algebra for $r=kcdot(n-1)$ with respect to the (Atiyah--Bott--)Goldman Poisson bracket and the swapping bracket. This is the building block of the general surface case. Two such injective Poisson algebra homomorphisms related to two ideal triangulations $mathcal{T}$ and $mathcal{T}$ are compatible with each other under the flips.
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.
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 symmetric functions in two parameters $t, theta$, which maps the $imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($imath$HL) functions. The (modified) $imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $imath$HL functions. The modified $imath$HL functions specialize at $theta=0$ to the modified HL functions; they specialize at $theta=1$ to the deformed universal characters of type C, which further specialize at $(t=0, theta =1)$ to the universal characters of type C.
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 have the corresponding central extension $S^3gl(n,H)oplus oplus_k (Ca_k)$. As a subalgebra of $S^3H$ we have the algebra $C[phi]$ of the Laurent polynomial spinors on $S^3$. Then we have a Lie subalgebra $hat{gl}(n, H)=C[phi] otimes gl(n, C)$ of $S^3gl(n,H)$, as well as its central extension by the 2-cocycles ${c_k}$ and the Euler vector field $d$: $hat{gl}=hat{gl}(n, H) oplus oplus_k(Ca_k)oplus Cd$ . The Lie algebra $hat{sl}(n,H)$ is defined as a Lie subalgebra of $hat{gl}(n,H)$ generated by $C[phi]otimes sl(n,C))$. We have the corresponding central extension of $hat{sl}(n,H)$ by the 2-cocycles ${c_k}$ and the derivation $d$, which becomes a Lie subalgebra $hat{sl}$ of $hat{gl}$. Let $h_0$ be a Cartan subalgebra of $sl(n,C)$ and $hat{h}=h_0 oplus oplus_k(Ca_k)oplus Cd$. The root space decomposition of the $ad(hat{h})$-representation of $hat{sl}$ is obtained. The set of roots is $Delta ={ m/2 delta + alpha ; alpha in Delta_0, m in Z} bigcup {m/2 delta ; m in Z }$ . And the root spaces are $hat{g}_{m/2 delta+ alpha}= C[phi ;m] otimes g_{alpha}$, for $alpha eq 0$ , $hat{g}_{m/2 delta}= C[phi ;m] otimes h_0$, for $m eq 0$, and $hat{g}_{0 delta}= hat{h}$, where $C[phi ;m]$ is the subspace with the homogeneous degree m. The Chevalley generators of $hat{sl}$ are given.
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