POISSON TRACES AND D-MODULES ON POISSON VARIETIESm
POISSON TRACES AND D-MODULES ON POISSON VARIETIESm
复制标题
柑橘微量元素和柑橘品种上的 D-微量元素m
DOI：
10.1007/s00039-010-0085-4
复制
发表时间：
2010-10-01
影响因子：
2.2
通讯作者：
Schedler, Travis
中科院分区：
文献类型：
--
作者：
Etingof, Pavel;Schedler, Travis
对于特征为零的代数闭域上的每一个泊松代数变量X，我们在X上正则地附加一个右d模M(X)。如果X是仿射的，则X上代数分布空间中的M(X)的解是X上的泊松迹，即哈密顿流下的不变分布。当X有有限多个辛叶时，证明M(X)是完整的。因此，当X是仿射且有有限多个辛叶时，X上的泊松迹空间是有限维的。更一般地说，对于任意态射phi: X - >y和任意拟相干泊松模N系，我们在X上附加了一个右d模M(phi) (X， N)，并证明了如果X有有限个辛叶，有限且N是相干的，则M（phi）是完整的。作为一个应用，我们推导出非交换滤除代数具有有限个不可约有限维表示，其相关的梯度代数在其中心上是有限的，其谱有有限多个辛叶。Ivan Losev在附录中加强了这一点，证明在这样的代数中，有有限多个素数理想，而且它们都是原始的。这包括辛反射代数。进一步，我们明确地描述了当X = V/G时，X上泊松轨迹的有限维空间（在仿射变体和紧C (a)-流形的设置下），其中V是辛的，G是忠实作用于V的有限群。
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(phi) (X, N) 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 (a)-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.