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To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are P...
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To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism φ: X → Y and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_φ(X,N) on X, and prove that it is holonomic if X has finitely many symplectic leaves, φ is finite, and N is coherent. As an application, we deduce that noncommutative filtered algebras, for which the associated graded algebra is finite over its center whose spectrum has finitely many symplectic leaves, have finitely many irreducible finite-dimensional representations. The appendix, by Ivan Losev, strengthens this to show that, in such algebras, there are finitely many prime ideals, and they are all primitive. This includes symplectic reflection algebras. Furthermore, we describe explicitly (in the settings of affine varieties and compact C~∞-manifolds) the finite-dimensional space of Poisson traces on X when X = V/G, where V is symplectic and G is a finite group acting faithfully on V.
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We introduce the notion of pseudo-Poisson Nijenhuis manifolds. These manifolds are generalizations of Poisson Nijenhuis manifolds defined by Magri and Morosi [13]. We show that there is a one-to-one correspondence between the pseu...
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We introduce the notion of pseudo-Poisson Nijenhuis manifolds. These manifolds are generalizations of Poisson Nijenhuis manifolds defined by Magri and Morosi [13]. We show that there is a one-to-one correspondence between the pseudo-Poisson Nijenhuis manifolds and certain quasi-Lie bialgebroid structures on the tangent bundle as in the case of Poisson Nijenhuis manifolds by Kosmann-Schwarzbach [7]. For that reason, we expand the general theory of the compatibility of a 2-vector field and a (1, 1)-tensor. We also introduce pseudo-symplectic Nijenhuis structures, and investigate properties of them. In particular, we show that those structures induce twisted Poisson structures [18].
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We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence wi...
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We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.
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? 2022 The AuthorsWe prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson...
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? 2022 The AuthorsWe prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson prime ideals between any two given Poisson prime ideals have the same length.
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Everyone has heard of the Poisson distribution, of Poisson brackets and of many other "Poisson something"s. But who was Poisson and what did he do to acquire such fame? At the initiative of its director, Brigitte Laude, the resear...
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Everyone has heard of the Poisson distribution, of Poisson brackets and of many other "Poisson something"s. But who was Poisson and what did he do to acquire such fame? At the initiative of its director, Brigitte Laude, the research library, "Mathématiques Informatique Recherche" (MIR), of the Université Pierre et Marie Curie in Paris, on the Jussieu campus, organised an exhibit "Siméon-Denis Poisson. Les mathématiques au service de la science", which was on display from 19 March to 19 June 2014 aiming to present some of Poisson's works and their modern continuation.
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The notion of characteristic pairs of Dirac structures was introduced by Liu in 2000. In this paper, the invariant Dirac structures on Poisson actions and pullback Dirac structures are characterized in terms of their characteristi...
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The notion of characteristic pairs of Dirac structures was introduced by Liu in 2000. In this paper, the invariant Dirac structures on Poisson actions and pullback Dirac structures are characterized in terms of their characteristic pairs. Poisson homogeneous spaces and Poisson reduction are discussed.
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Let A be a Poisson algebra over a field k with characteristic zero, let gamma, alpha be Poisson derivations on A such that gamma alpha = a gamma and 0 not equal rho is an element of k. Here the notion of a gamma-Poisson normal ele...
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Let A be a Poisson algebra over a field k with characteristic zero, let gamma, alpha be Poisson derivations on A such that gamma alpha = a gamma and 0 not equal rho is an element of k. Here the notion of a gamma-Poisson normal element is introduced, it is proved that the polynomial algebra A[y, x] has a Poisson structure defined by {y, a} = a(a)y, {x, a} = beta(a)x, {x, y} = beta(y)x + delta(y) for a is an element of A, where beta is a Poisson derivation on A[y] defined by beta|(A) =gamma - alpha, beta(y) = rho y and delta is a derivation on A[y] such that delta|(A) = 0, and its Poisson simplicity criterion is established and endorsed by examples.
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We establish the concept of a quotient affine Poisson group, and study the reduced Poisson action of the quotient of an affine poisson group G on the quotient of an affine Poisson G-variety V. The Poisson morphisms (including equi...
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We establish the concept of a quotient affine Poisson group, and study the reduced Poisson action of the quotient of an affine poisson group G on the quotient of an affine Poisson G-variety V. The Poisson morphisms (including equivariant cases) between Poisson affine varieties are also discussed.
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The Poisson distribution is a benchmark for modeling count data. Its equidispersion constraint, however, does not accurately represent real data. Most real datasets express overdispersion; hence attention in the statistics communi...
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The Poisson distribution is a benchmark for modeling count data. Its equidispersion constraint, however, does not accurately represent real data. Most real datasets express overdispersion; hence attention in the statistics community focuses on associated issues. More examples are surfacing, however, that display underdispersion, warranting the need to highlight this phenomenon and bring more attention to those models that can better describe such data structures. This work addresses various sources of data underdispersion and surveys several distributions that can model underdispersed data, comparing their performance on applied datasets.
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We describe an implementation to solve Poissons equation for an isolated system on a unigrid mesh using FFTs. The method solves the equation globally on mesh blocks distributed across multiple processes on a distributed-memory par...
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We describe an implementation to solve Poissons equation for an isolated system on a unigrid mesh using FFTs. The method solves the equation globally on mesh blocks distributed across multiple processes on a distributed-memory parallel computer. Test results to demonstrate the convergence and scaling properties of the implementation are presented. The solver is offered to interested users as the library PSPFFT. Program summary: Program title: PSPFFT Catalogue identifier: AEJK-v1-0 Program summary URL: http://cpc.cs.qub.ac.uk/ summaries/AEJK-v1-0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 110 243 No. of bytes in distributed program, including test data, etc.: 16 332 181 Distribution format: tar.gz Programming language: Fortran 95 Computer: Any architecture with a Fortran 95 compiler, distributed memory clusters Operating system: Linux, Unix Has the code been vectorized or parallelized?: Yes, using MPI. An arbitrary number of processors may be used (subject to some constraints). The program has been tested on from 1 up to ～ 13 000 processors. RAM: Depends on the problem size, approximately 170 MBytes for 48~3 cells per process. Classification: 4.3, 6.5 External routines: MPI (http://www.mcs.anl.gov/mpi/), FFTW (http://www.fftw.org), Silo (https://wci.llnl.gov/codes/silo/) (only necessary for running test problem). Nature of problem: Solving Poissons equation globally on unigrid mesh distributed across multiple processes on distributed memory system. Solution method: Numerical solution using multidimensional discrete Fourier Transform in a parallel Fortran 95 code. Unusual features: This code can be compiled as a library to be readily linked and used as a blackbox Poisson solver with other codes. Running time: Depends on the size of the problem, but typically less than 1 second per solve.
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