Poisson geometry of PI three‐dimensional Sklyanin algebras

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 Z 3 × 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 .