References of "Sun, Zhe 50031864"
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See detailRank n swapping algebra for PGLn Fock--Goncharov X moduli space
Sun, Zhe UL

in Mathematische Annalen (in press)

The rank $n$ swapping multifraction algebra is a field of cross ratios up to $(n+1)\times (n+1)$-determinant relations equipped with a Poisson bracket, called the {\em swapping bracket}, defined on the ... [more ▼]

The rank $n$ swapping multifraction algebra is a field of cross ratios up to $(n+1)\times (n+1)$-determinant relations equipped with a Poisson bracket, called the {\em swapping bracket}, defined on the set of ordered pairs of points of a circle using linking numbers. Let $D_k$ be a disk with $k$ points on its boundary. The moduli space $\mathcal{X}_{\operatorname{PGL}_n,D_k}$ is the building block of the Fock--Goncharov $\mathcal{X}$ moduli space for any general surface. Given any ideal triangulation of $D_k$, we find an injective Poisson algebra homomorphism from the rank $n$ Fock--Goncharov algebra for $\mathcal{X}_{\operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifraction algebra with respect to the Atiyah--Bott--Goldman Poisson bracket and the swapping bracket. Two such injective Poisson algebra homomorphisms related to two ideal triangulations $\mathcal{T}$ and $\mathcal{T}'$ are compatible with each other under the flips. [less ▲]

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See detailFlows on the PSL(V)-Hitchin component
Sun, Zhe UL; Wienhard, Anna; Zhang, Tengren

in Geometric and Functional Analysis (2020), 30

In this article we define new flows on the Hitchin components for PSL(n,R). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other ... [more ▼]

In this article we define new flows on the Hitchin components for PSL(n,R). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n = 2. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the PGL(V)-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component. [less ▲]

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See detailRank n swapping algebra for Grassmannian
Sun, Zhe UL

E-print/Working paper (2019)

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 (K^n×K^{n∗})^r/GL(n,K) is its geometric mode. In this ... [more ▼]

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 (K^n×K^{n∗})^r/GL(n,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. [less ▲]

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See detailThe Goldman symplectic form on the PSL(V)-Hitchin component
Sun, Zhe UL; Zhang, Tengren

E-print/Working paper (2017)

This article is the second of a pair of articles about the Goldman symplectic form on the PSL(V )-Hitchin component. We show that any ideal triangulation on a closed connected surface of genus at least 2 ... [more ▼]

This article is the second of a pair of articles about the Goldman symplectic form on the PSL(V )-Hitchin component. We show that any ideal triangulation on a closed connected surface of genus at least 2, and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the Hitchin component. Using this, we prove that a large class of flows defined in the companion paper [SWZ17] are Hamiltonian. We also construct an explicit collection of Hamiltonian vector fields on the Hitchin component that give a symplectic basis at every point. These are used in the companion paper to compute explicit global Darboux coordinates for the Hitchin component. [less ▲]

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See detailMcShane identities for Higher Teichmüller theory and the Goncharov-Shen potential
Sun, Zhe UL; Huang, Yi

in Memoirs of the American Mathematical Society (n.d.)

We derive generalizations of McShane's identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which ... [more ▼]

We derive generalizations of McShane's identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric. [less ▲]

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