## Abstract

We give the three-dimensional Sklyanin algebras S that are module-finite over their center Z, the structure of a Poisson Z-order (in the sense of Brown–Gordon). We show that the induced Poisson bracket on Z is nonvanishing and is induced by an explicit potential. The Z3 × k× -orbits of symplectic cores of the Poisson structure are determined (where the group acts on S by algebra automorphisms). In turn, this is used to analyze the finite-dimensional quotients of S by central annihilators: there are three distinct isomorphism classes of such quotients in the case (n, 3) = 1 and two in the case (n, 3) = 1, where n is the order of the elliptic curve automorphism associated to S. The Azumaya locus of S is determined, extending results of Walton for the case n, 3 = 1.

Original language | English (US) |
---|---|

Pages (from-to) | 1471-1500 |

Number of pages | 30 |

Journal | Proceedings of the London Mathematical Society |

Volume | 118 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2019 |

## Keywords

- 14A22 (primary)
- 16G30
- 17B63
- 81S10 (secondary)

## ASJC Scopus subject areas

- General Mathematics