Center of Poisson and Skein Algebras Associated with Loops on Surfaces

We discuss and develop a systematic method to compute the Poisson center (Casimir) of various Poisson algebras associated with loops on orientable surfaces (possibly with boundary and punctures) introduced by Goldman and Wolpert in the 1980 while studying Thurston’s earthquakes deformations. Our computation extends a result of Etingof to all finite-type hyperbolic surfaces. We use these methods to compute the center of various skein algebras introduced by Turaev for the quantization of these Poisson algebras. As another application of our results, we compute the center of homotopy skein algebra introduced by Hoste and Przytycki.