Poisson orders, symplectic reflection algebras and representation theory

By Kenneth A. Brown, I. Gordon. 2002

137 citations recorded by the source corpus.

We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in positive characteristic. Quite generally, we study this class of algebras from the point of view of Poisson geometry, exhibiting connections between their representation theory and some well-known geometric constructions. As an application, we employ our results in the study of symplectic reflection algebras, completing work of Etingof and Ginzburg on when these algebras are finite over their centres, and providing a framework for the study of their representation theory in the latter case.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa6371a7fa5434309ef1c14e351309cf71eef07b

How researchers see this paper

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In Algebra and Number Theory

1. K. Goodearl
2. Kenneth A. Brown
3. V. Ginzburg
4. I. Losev
5. James J. Zhang
6. A. Premet
7. I. Gordon
8. T. Arakawa

In Geometry and Topology

1. Kenneth A. Brown
2. K. Goodearl
3. V. Ginzburg
4. James J. Zhang
5. I. Gordon
6. I. Losev
7. Michel van den Bergh
8. Pavel Etingof

In Mathematical Physics

1. Kenneth A. Brown
2. K. Goodearl
3. V. Ginzburg
4. I. Losev
5. A. Premet
6. I. Gordon
7. Pavel Etingof
8. T. Arakawa

Citations in the corpus

• Symplectic Reflection Algebras (2003, background): “The present version, due to Brown and Gordon [4], avoids those hypotheses, but does require that the leaves are algebraic.”
• LECTURES ON NAKAJIMA'S QUIVER VARIETIES (2009, background): “(iii) The Poisson variety X is a union of finitely many symplectic leaves X = tX , [K4], and each symplectic leaf X is a locally closed smooth algebraic subvariety of X, [ BG ].”
• A Dixmier-Moeglin equivalence for Poisson algebras with torus actions (2005, methodology): “The ideals I of R for which {R, I} ⊆ I are called Poisson ideals , and within each ideal J of R there is a largest Poisson ideal, which we call (following [3]) the Poisson core of J .”