Poisson geometry of PI 3-dimensional Sklyanin algebras

Abstract

We give the 3-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 non-vanishing and is induced by an explicit potential. The ${\mathbb{Z}}_3\times {\mathbb{k}}^{\times }$-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 3 distinct isomorphism classes of such quotients in the case $(n,3)\ne 1$ and 2 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 $(n,3)=1$.

v3: 29 pages + references. Includes changes per the referee's suggestions and other minor changes. To appear in the Proceedings of the London Mathematical Society

References in corpus (3)

• The geometry of representations of 3-dimensional Sklyanin algebras
• Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2
• The irreducible representations of 3-dimensional Sklyanin algebras

Cited by in corpus (8)

• The Zariski cancellation problem for Poisson algebras
• Reflexive hull discriminants and applications
• Reflection Groups and Rigidity of Quadratic Poisson Algebras
• Poisson Dixmier-Moeglin equivalence from a topological point of view
• Root of unity quantum cluster algebras and discriminants
• The irreducible representations of 3-dimensional Sklyanin algebras
• The superpotential $XYZ+XZY+\frac{c}{3}(X^3+Y^3+Z^3)$
• Derived equivalences for a class of PI algebras