Symplectic reflection algebras in positive characteristic as Ore extensions

Abstract

We investigate PBW deformations ${\mathsf{H}}_{\lambda }$ of $k[x,y]⋊G$ where G is the cyclic group of order p and the ground field k has characteristic p. The algebras ${\mathsf{H}}_{\lambda }$ are a version of symplectic reflection algebras that only exist in positive characteristic. They also admit a presentation as Ore extensions over $k[x]\otimes kG,$ and the combinatorics of the derivation used in this presentation is related to André and Eulerian polynomials. We find the center of ${\mathsf{H}}_{\lambda }$ and classify the simple modules. We also study a version of ${\mathsf{H}}_{\lambda }$ in which G is replaced by an elementary abelian p-group G^r.

Keywords:

• Skew group algebras
• quadratic algebras
• Ore extensions
• deformations
• symplectic reflection algebras
• elementary abelian p-groups
• André polynomials
• modular representations in defining characteristic

2010 Mathematics Subject Classification:

• 16S32
• 16S35
• 16S37
• 16S80
• 16W70
• 16G99

Acknowledgements

I would like to thank Victor Ginzburg for suggesting the problem and for useful conversations, and the anonymous referee for suggesting many improvements to the text.

Notes

1 See [9, Section 2.5], where the statement is attributed to Dixmier.