Relevant bibliographies by topics / Symplectic groupoids

Academic literature on the topic 'Symplectic groupoids'

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Published: 8 February 2024

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Journal articles on the topic "Symplectic groupoids"

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MACKENZIE, K. C. H. "ON SYMPLECTIC DOUBLE GROUPOIDS AND THE DUALITY OF POISSON GROUPOIDS." International Journal of Mathematics 10, no. 04 (June 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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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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XU, PING. "ON POISSON GROUPOIDS." International Journal of Mathematics 06, no. 01 (February 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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Ševera, Pavol, and Michal Širaň. "Integration of Differential Graded Manifolds." International Mathematics Research Notices 2020, no. 20 (February 15, 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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Cattaneo, Alberto S., and Ivan Contreras. "Relational Symplectic Groupoids." Letters in Mathematical Physics 105, no. 5 (April 22, 2015): 723–67. http://dx.doi.org/10.1007/s11005-015-0760-3.

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Gualtieri, Marco, and Songhao Li. "Symplectic Groupoids of Log Symplectic Manifolds." International Mathematics Research Notices 2014, no. 11 (March 1, 2013): 3022–74. http://dx.doi.org/10.1093/imrn/rnt024.

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Mehta, Rajan Amit, and Xiang Tang. "Constant symplectic 2-groupoids." Letters in Mathematical Physics 108, no. 5 (November 15, 2017): 1203–23. http://dx.doi.org/10.1007/s11005-017-1026-z.

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

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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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Dissertations / Theses on the topic "Symplectic groupoids"

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Cosserat, Oscar. "Theory and Construction of Structure Preserving Integrators in Poisson Geometry." Electronic Thesis or Diss., La Rochelle, 2023. http://www.theses.fr/2023LAROS018.

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Nous introduisons pour toute structure de Poisson sur une variété la notion de bi-réalisation et l'illustrons par des exemples. Nous définissons les intégrateurs de Poisson hamiltoniens comme des intégrateurs de Poisson dont la trajectoire discrète suit le flot d'un hamiltonien dépendant du temps. Ensuite, une construction d'intégrateur de Poisson hamiltonien pour une structure de Poisson, un Hamiltonien H, un ordre k et un pas de temps t quelconques est donnée via une troncature à l'ordre k de la transformée de Hamilton-Jacobi S¬t(H) de H sur une bi-réalisation de la structure de Poisson. Nous définissons aussi la suite de Farmer et expliquons comment elle permet de résoudre explicitement l'équation de Hamilton-Jacobi à un ordre arbitraire. Nous expliquons comment les groupoïdes symplectiques locaux fournissent une interprétation géométrique de la notion de bi-réalisation. Nous définissons pour tout hamiltonian dépendant du temps H sa série de Magnus, pour construire pour tout intégrateur hamiltonien de Poisson un hamiltonien modifié. En conclusion, nous comparons nos intégrateurs avec des méthodes de Runge-Kutta sur les exemples du solide rigide et des équations différentielles de Lodka-Volterra, en particulier concernant leur comportement à long terme. En géométrie de Dirac, nous introduisons le 2-cocyle horizontal canonique d'une structure de Dirac. Sous la condition suffisante de son exactitude, nous exhibons pour tout hamiltonien H une fonctionnelle pour laquelle les points critiques sont exactement les courbes intégrales des champs de vecteurs hamiltoniens de H. Nous déduisons aussi du résultat précédent une généralisation de la transformée de Legendre aux structures de Dirac
We introduce for any Poisson structure on a manifold the notion of bi-realisation and illustrate it by examples. We define Hamiltonian Poisson integrators as Poisson integrators for which discrete trajectory follows the flow of a time-dependent Hamiltonian. Next, a construction of a Hamiltonian Poisson integrator for generic Poisson structure, Hamiltonian H, order k and time-step t are given via any truncation at order k of the Hamilton-Jacobi transform S¬t(H) of the Hamiltonian H on a bi-realisation of the Poisson structure. We also define the Farmer sequence and we explain how it gives explicit recursive formulae to solve Hamilton-Jacobi equation at an arbitrary order. We explain how local symplectic groupoids provide a geometric interpretation of the notion of bi-realisation. We define for any time-dependent Hamiltonian H its Magnus series to construct, for any Hamiltonian Poisson integrator, a modified Hamiltonian. To conclude, we compare our integrators with Runge-Kutta methods on the example of rigid body dynamics and Lotka-Volterra differential equations, in particular on long run simulations. In Dirac geometry, we introduce the canonical horizontal 2-cocycle of a Dirac structure. Under the sufficiency condition of its exactness, we exhibit for any Hamiltonian H a functional for which critical points are exactly integral curves of Hamiltonian vector fields of H. We also deduce from the previous result a generalisation of the Legendre transform to Dirac structures
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Li, Travis Songhao. "Constructions of Lie Groupoids." Thesis, 2013. http://hdl.handle.net/1807/43638.

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In this thesis, we develop two methods for constructing Lie groupoids. The first method is a blow-up construction, corresponding to the elementary modification of a Lie algebroid along a subalgebroid over some closed hypersurface. This construction may be specialized to the Poisson groupoids and Lie bialgebroids. We then apply this method to three cases. The first is the adjoint Lie groupoid integrating the Lie algebroid of vector fields tangent to a collection of normal crossing hypersurfaces. The second is the adjoint symplectic groupoid of a log symplectic manifold. The third is the adjoint Lie groupoid integrating the tangent algebroid of a Riemann surface twisted by a divisor. The second method is a gluing construction, whereby Lie groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to construct and classify the Lie groupoids integrating the given Lie algebroid. We apply this method to the aforementioned cases, albeit with small differences, and characterize the category of integrations in each case.
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Books on the topic "Symplectic groupoids"

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Dazord, Pierre, and Alan Weinstein, eds. Symplectic Geometry, Groupoids, and Integrable Systems. New York, NY: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4613-9719-9.

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Séminaire, sud-rhodanien de géométrie (6th 1989 Berkeley Calif ). Symplectic geometry, groupoids, and integrable systems. New York: Springer-Verlag, 1991.

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Séminaire Sud-Rhodanien de Géométrie (6th 1989 Berkeley, Calif.). Symplectic geometry, groupoids, and integrable systems: Séminaire Sud Rhodanien de Géométrie à Berkeley (1989). Edited by Dazord P and Weinstein Alan. New York: Springer-Verlag, 1991.

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Gekhtman, Michael. Cluster algebra and Poisson geometry. Providence, R.I: American Mathematical Society, 2010.

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(Editor), Pierre Dazord, and Alan Weinstein (Editor), eds. Symplectic Geometry, Groupoids, and Integrable Systems: Seminaire Sud Rhodanien de Geometrie a Berkeley (1989) (Mathematical Sciences Research Institute Publications). Springer, 1991.

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Weinstein, Alan, and Pierre Dazord. Symplectic Geometry, Groupoids, and Integrable Systems: Séminaire Sud Rhodanien de Géométrie à Berkeley. Springer, 2012.

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Weinstein, Alan, and Pierre Dazord. Symplectic Geometry, Groupoids, and Integrable Systems: Séminaire Sud Rhodanien de Géométrie à Berkeley. Springer, 2012.

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Crainic, Marius, Rui Loja Fernandes, and Ioan Marcut. Lectures on Poisson Geometry. American Mathematical Society, 2021.

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Lectures on Poisson Geometry. American Mathematical Society, 2021.

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Book chapters on the topic "Symplectic groupoids"

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Xu, Ping. "Morita Equivalent Symplectic Groupoids." In Mathematical Sciences Research Institute Publications, 291–311. New York, NY: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4613-9719-9_20.

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Cattaneo, Alberto S., and Giovanni Felder. "Poisson sigma models and symplectic groupoids." In Quantization of Singular Symplectic Quotients, 61–93. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8364-1_4.

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Vaisman, Izu. "Realizations of Poisson Manifolds by Symplectic Groupoids." In Lectures on the Geometry of Poisson Manifolds, 135–59. Basel: Birkhäuser Basel, 1994. http://dx.doi.org/10.1007/978-3-0348-8495-2_10.

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Weinstein, Alan. "Symplectic Groupoids, Geometric Quantization, and Irrational Rotation Algebras." In Mathematical Sciences Research Institute Publications, 281–90. New York, NY: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4613-9719-9_19.

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Lauter, Robert, and Victor Nistor. "Analysis of geometric operators on open manifolds: A groupoid approach." In Quantization of Singular Symplectic Quotients, 181–229. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8364-1_8.

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"Symplectic groupoids." In Lectures on Poisson Geometry, 361–418. Providence, Rhode Island: American Mathematical Society, 2021. http://dx.doi.org/10.1090/gsm/217/17.

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Marle, C. M. "Lie, Symplectic, and Poisson Groupoids and Their Lie Algebroids." In Encyclopedia of Mathematical Physics, 312–20. Elsevier, 2006. http://dx.doi.org/10.1016/b0-12-512666-2/00145-0.

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"Poisson and Symplecfie Groupoids." In General Theory of Lie Groupoids and Lie Algebroids, 408–45. Cambridge University Press, 2005. http://dx.doi.org/10.1017/cbo9781107325883.015.

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Conference papers on the topic "Symplectic groupoids"

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Mackenzie, Kirill. "FROM SYMPLECTIC GROUPOIDS TO DOUBLE STRUCTURES." In Villa de Leyva Summer School. WORLD SCIENTIFIC, 2016. http://dx.doi.org/10.1142/9789814730884_0005.

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Bonechi, Francesco, Nicola Ciccoli, and Marco Tarlini. "Quantization of the symplectic groupoid." In Proceedings of the Corfu Summer Institute 2011. Trieste, Italy: Sissa Medialab, 2012. http://dx.doi.org/10.22323/1.155.0060.

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