Poisson geometry and representations of PI 4-dimensional Sklyanin algebras

TL;DR

This paper constructs a Poisson structure on 4-dimensional Sklyanin algebras, classifies their irreducible representations using geometric and algebraic methods, and analyzes their singularities and discriminant ideals.

Contribution

It introduces a Poisson order structure on Sklyanin algebras and combines geometric, algebraic, and representation-theoretic techniques for classification.

Findings

Constructed a non-vanishing Poisson bracket on the center of S.

Classified irreducible representations using geometric and algebraic methods.

Described the singular locus and zero sets of discriminant ideals.

Abstract

Take S to be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its center Z; thus, S is PI. Our first result is the construction of a Poisson Z-order structure on S such that the induced Poisson bracket on Z is non-vanishing. We also provide the explicit Jacobian structure of this bracket, leading to a description of the symplectic core decomposition of the maximal spectrum Y of Z. We then classify the irreducible representations of S by combining (1) the geometry of the Poisson order structures, with (2) algebro-geometric methods for the elliptic curve attached to S, along with (3) representation-theoretic methods using line and fat point modules of S. Along the way, we improve results of Smith and Tate obtaining a description the singular locus of Y for such S. The classification results for irreducible representations are in turn used to determine the zero sets of the discriminants ideals of these algebras S.