POISSON TRACES AND D-MODULES ON POISSON VARIETIES
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 flo...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Springer
2011
|
| Online Access: | http://hdl.handle.net/1721.1/62847 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826215945996075008 |
|---|---|
| author | Etingof, Pavel I. Schedler, Travis |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I. Schedler, Travis |
| author_sort | Etingof, Pavel I. |
| collection | MIT |
| description | 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 [phi]: X → Y and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M [subscript phi] (X,N)M(XN) on X, and prove that it is holonomic if X has finitely many symplectic leaves, [phi] 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 [C superscript infinity symbol -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. |
| first_indexed | 2024-09-23T16:39:58Z |
| format | Article |
| id | mit-1721.1/62847 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:39:58Z |
| publishDate | 2011 |
| publisher | Springer |
| record_format | dspace |
| spelling | mit-1721.1/628472022-10-03T07:27:53Z POISSON TRACES AND D-MODULES ON POISSON VARIETIES Etingof, Pavel I. Schedler, Travis Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I. Etingof, Pavel I. Schedler, Travis 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 [phi]: X → Y and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M [subscript phi] (X,N)M(XN) on X, and prove that it is holonomic if X has finitely many symplectic leaves, [phi] 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 [C superscript infinity symbol -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. 2011-05-19T19:41:07Z 2011-05-19T19:41:07Z 2010-10 Article http://purl.org/eprint/type/JournalArticle 1016-443X http://hdl.handle.net/1721.1/62847 Etingof, Pavel, and Travis Schedler. “Poisson Traces and D-Modules on Poisson Varieties.” Geometric And Functional Analysis 20.4 (2010) : 958-987-987. Copyright © 2010, Springer https://orcid.org/0000-0002-0710-1416 en_US http://dx.doi.org/10.1007/s00039-010-0085-4 Geometric and Functional Analysis Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Springer Prof. Etingof via Michael Noga (arXiv ms.) |
| spellingShingle | Etingof, Pavel I. Schedler, Travis POISSON TRACES AND D-MODULES ON POISSON VARIETIES |
| title | POISSON TRACES AND D-MODULES ON POISSON VARIETIES |
| title_full | POISSON TRACES AND D-MODULES ON POISSON VARIETIES |
| title_fullStr | POISSON TRACES AND D-MODULES ON POISSON VARIETIES |
| title_full_unstemmed | POISSON TRACES AND D-MODULES ON POISSON VARIETIES |
| title_short | POISSON TRACES AND D-MODULES ON POISSON VARIETIES |
| title_sort | poisson traces and d modules on poisson varieties |
| url | http://hdl.handle.net/1721.1/62847 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT etingofpaveli poissontracesanddmodulesonpoissonvarieties AT schedlertravis poissontracesanddmodulesonpoissonvarieties |