Center of Poisson and skein algebras associated to loops on surfaces

TL;DR

This paper develops a systematic method to compute the Poisson center of algebras related to loops on surfaces, extending previous results and applying to skein algebras for quantization and deformation studies.

Contribution

It introduces a unified approach to compute centers of Poisson and skein algebras associated with surface loops, extending existing results to all finite type hyperbolic surfaces.

Findings

Computed the Poisson center for various surface-associated algebras.

Extended Etingof's result to all finite type hyperbolic surfaces.

Determined the centers of skein and homotopy skein algebras.

Abstract

We discuss and develop a systematic method to compute the Poisson center (Casimir) of various Poisson algebras associated to loops on orientable surfaces (possibly with boundary and punctures) introduced by Goldman and Wolpert in 80's 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.