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Relevant bibliographies by topics / Symplectic groupoids

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Journal articles

Academic literature on the topic 'Symplectic groupoids'

Author: Grafiati

Published: 8 February 2024

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Journal articles on the topic "Symplectic groupoids"

1

MACKENZIE, K. C. H. "ON SYMPLECTIC DOUBLE GROUPOIDS AND THE DUALITY OF POISSON GROUPOIDS." International Journal of Mathematics 10, no. 04 (1999): 435–56. http://dx.doi.org/10.1142/s0129167x99000185.

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We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical isomorphisms for tangent and cotangent structures of the author and Ping Xu.

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2

Cattaneo, Alberto S., Benoit Dherin, and Giovanni Felder. "Formal Lagrangian Operad." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–36. http://dx.doi.org/10.1155/2010/643605.

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Given a symplectic manifoldM, we may define an operad structure on the the spacesOkof the Lagrangian submanifolds of(M¯)k×Mvia symplectic reduction. IfMis also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semiclassical part of Kontsevich's deformation ofC∞(ℝd) is a deformation of the trivial symplectic groupoid structure ofT∗ℝd.

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3

XU, PING. "ON POISSON GROUPOIDS." International Journal of Mathematics 06, no. 01 (1995): 101–24. http://dx.doi.org/10.1142/s0129167x95000080.

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Some important properties of Poisson groupoids are discussed. In particular, we obtain a useful formula for the Poisson tensor of an arbitrary Poisson groupoid, which generalizes the well-known multiplicativity condition for Poisson groups. Morphisms between Poisson groupoids and between Lie bialgebroids are also discussed. In particular, for a special class of Lie bialgebroid morphisms, we give an explicit lifting construction. As an application, we prove that a Poisson group action on a Poisson manifold lifts to a Poisson action on its α-simply connected symplectic groupoid.

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4

Ševera, Pavol, and Michal Širaň. "Integration of Differential Graded Manifolds." International Mathematics Research Notices 2020, no. 20 (2019): 6769–814. http://dx.doi.org/10.1093/imrn/rnz004.

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Abstract We consider the problem of integration of $L_\infty $-algebroids (differential non-negatively graded manifolds) to $L_\infty $-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer–Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer–Cartan equation to closed differential forms. Following the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold is (the nerve of) a local Lie $\ell $-groupoid. The gauge condition can be imposed only locally in the base of the $L_\infty $-algebroid; the resulting local $\ell $-groupoids glue up to a coherent homotopy, that is, we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local $\ell $-groupoids. Finally, we show that a $k$-symplectic differential non-negatively graded manifold integrates to a local $k$-symplectic Lie $\ell$-groupoid; globally, these assemble to form an $A_\infty$-functor. As a particular case for $k=2$, we obtain integration of Courant algebroids.

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5

Cattaneo, Alberto S., and Ivan Contreras. "Relational Symplectic Groupoids." Letters in Mathematical Physics 105, no. 5 (2015): 723–67. http://dx.doi.org/10.1007/s11005-015-0760-3.

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6

Gualtieri, Marco, and Songhao Li. "Symplectic Groupoids of Log Symplectic Manifolds." International Mathematics Research Notices 2014, no. 11 (2013): 3022–74. http://dx.doi.org/10.1093/imrn/rnt024.

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7

Mehta, Rajan Amit, and Xiang Tang. "Constant symplectic 2-groupoids." Letters in Mathematical Physics 108, no. 5 (2017): 1203–23. http://dx.doi.org/10.1007/s11005-017-1026-z.

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8

戴, 远莉. "Symplectic Reduction for Cotangent Groupoids." Pure Mathematics 11, no. 03 (2021): 323–29. http://dx.doi.org/10.12677/pm.2021.113043.

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9

Weinstein, Alan. "Symplectic groupoids and Poisson manifolds." Bulletin of the American Mathematical Society 16, no. 1 (1987): 101–5. http://dx.doi.org/10.1090/s0273-0979-1987-15473-5.

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10

Li, Songhao, and Dylan Rupel. "Symplectic groupoids for cluster manifolds." Journal of Geometry and Physics 154 (August 2020): 103688. http://dx.doi.org/10.1016/j.geomphys.2020.103688.

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