Symplectic groupoids for cluster manifolds

Li, Songhao; Rupel, Dylan

doi:10.1016/j.geomphys.2020.103688

Cited by 7 publications

(7 citation statements)

References 37 publications

(56 reference statements)

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“…and since multiplication in D induces a local isomorphism between G * × G and D, the first relation follows from (8). The second relation is similarly proved.…”

Section: Local Dressing Actionsmentioning

confidence: 63%

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Mouquin

2021

Journal of Symplectic Geometry

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Given a standard complex semisimple Poisson Lie group (G, π st ), generalised double Bruhat cells G u,v and generalised Bruhat cells O u equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang-Hua Lu and the author. We prove in this paper that G u,u is naturally a Poisson groupoid over O u , extending a result from the aforementioned authors about double Bruhat cells in (G, π st ).Our result on G u,u is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to "twist" a direct product of Poisson groupoids. Victor Mouquin 7 Generalised Double Bruhat cells 1023 8 Poisson action of a double symplectic groupoid on generalised double Bruhat cells 1032 9 The Poisson groupoid (G w, w, π w, w) 1036 References 1044

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“…and since multiplication in D induces a local isomorphism between G * × G and D, the first relation follows from (8). The second relation is similarly proved.…”

Section: Local Dressing Actionsmentioning

confidence: 63%

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Mouquin

2021

Journal of Symplectic Geometry

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“…with (a ij ) i,j a skew-symmetric matrix is computed in [16] and is shown to be globally diffeomorphic to T * R n . The explicit structures given in [16] can be modified such that G = T * R n is equipped with the canonical symplectic structure. The source and target maps defined in [16] then become, with (x, p) cotangent coordinates on T * R n :…”

Section: Examples Of Poisson Integratorsmentioning

confidence: 99%

Symplectic Groupoids for Poisson Integrators

Cosserat¹

2022

Preprint

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We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators.

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“…In their setting the deformation of the toric gluing is obtained using Hamiltonian flows of the groupoid charts. It is a very interesting problem to explore the relation of our approach and that of [LR20].…”

Section: An Examplementioning

confidence: 99%

“…We would like to stress that -mutation formulas with principal coefficients can be obtained using the approach of [LR20] where the authors construct symplectic groupoids integrating log-canonical Poisson structures on -cluster varieties and their special completions. In their setting the deformation of the toric gluing is obtained using Hamiltonian flows of the groupoid charts.…”

Section: Introductionmentioning

confidence: 99%

Toric degenerations of cluster varieties and cluster duality

Bossinger¹,

Frías-Medina²,

Magee³

et al. 2020

Compositio Math.

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We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

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Symplectic groupoids for cluster manifolds

Cited by 7 publications

References 37 publications

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Symplectic Groupoids for Poisson Integrators

Toric degenerations of cluster varieties and cluster duality

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