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1. The partial compactification of the universal centralizer

   https://doi.org/10.1007/s00029-023-00873-8  

   Bălibanu, Ana (November 2023, Selecta Mathematica)

   The universal centralizer of a semisimple algebraic group is the family of centralizers of regular elements, parametrized by their conjugacy classes. When the group is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification of the universal centralizer by taking the closure of each fiber in the wonderful compactification. We use the geometry of the wonderful compactification to give an explicit description of the symplectic leaves of this new space. We also show that its compactified centralizer fibers are isomorphic to certain Hessenberg varieties—we apply this connection to compute the singular cohomology of the compactification, and to study the geometry of the corresponding universal Hessenberg family. 

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2. Poisson manifolds of compact types (PMCT 1)

   https://doi.org/10.1515/crelle-2017-0006  

   Crainic, Marius; Fernandes, Rui Loja; Martínez Torres, David (November 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompassesseveral classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamentalproperties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like thede Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.)and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTsare related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivialPMCT related to a classical question on the topology of the orbits of a free symplectic circle action.In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry,Hamiltonian G -spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes. 

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3. Feigin–Odesskii brackets associated with Kodaira cycles and positroid varieties

   https://doi.org/10.1112/plms.70054  

   Hua, Zheng; Polishchuk, Alexander (May 2025, Proceedings of the London Mathematical Society)

   Abstract We establish a link between open positroid varieties in the Grassmannians and certain moduli spaces of complexes of vector bundles over Kodaira cycle , using the shifted Poisson structure on the latter moduli spaces and relating them to the standard Poisson structure on . This link allows us to solve a classification problem for extensions of vector bundles over . Based on this solution we further classify the symplectic leaves of all positroid varieties in with respect to the standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of with the standard Poisson structure as an open substack of the stack of vector bundles on . 

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4. Reduction of symmetries of simple hybrid mechanical systems,

   Irazu, M; Colombo, L; Bloch, A (December 2021, IFACPapersOnLine)

   Abstract: This paper investigates reduction by symmetries in simple hybrid mechanical systems, in particular, symplectic and Poisson reduction for simple hybrid Hamiltonian and Lagrangian systems. We give general conditions for whether it is possible to perform a symplectic reduction for simple hybrid Lagrangian system under a Lie group of symmetries and we also provide sufficient conditions for perform 

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5. Poisson Trace Orders

   https://doi.org/10.1093/imrn/rnad086  

   Brown, Ken; Yakimov, Milen (May 2023, International Mathematics Research Notices)

   Abstract The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras. 

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