Poisson algebras of curves on bordered surfaces and skein quantization

TL;DR

This paper introduces a (co-)Poisson algebra structure on curves on bordered surfaces, generalizing Goldman and Turaev brackets, and demonstrates its quantization aligns with Muller's skein algebra, bridging geometric and algebraic frameworks.

Contribution

It defines a new (co-)Poisson algebra for curves on bordered surfaces and establishes its quantization as the skein algebra, extending previous algebraic structures in surface topology.

Findings

Defined a (co-)Poisson (co)bracket on curves

Established a Poisson algebra for unoriented curves

Proved quantization matches Muller's skein algebra

Abstract

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quotient of the vector space spanned by the regular homotopy classes of curves on the bordered surface by generalizing the Goldman bracket and the Turaev cobracket. Moreover, we define a Poisson algebra of unoriented curves on a bordered surface and show that a quantization of the Poisson algebra coincides with the skein algebra of the bordered surface defined by Muller.