Finite-dimensional Simple Poisson Modules

We classify the finite-dimensional simple Poisson modules for two Poisson algebras. The first is related to the invariants for an automorphism of the torus and to the cyclically q-deformed algebra U-q' (so(3)) of [1,2]. We find that there are five d-dimensional simple Poisson modules for each d >= 1. The second is the Poisson algebra arising from the quantized enveloping algebra U-q(sl(2)) using a presentation discovered by Ito, Terwilliger and Weng [3] and we find that there are two d-dimensional simple Poisson modules for each d >= 1.