Poisson algebras of curves on bordered surfaces and skein quantization

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.

arXiv comment23 pages, 9 figures; v2: references added; v3: references added, minor changes

References in corpus (4)

• A positive basis for surface skein algebras
• Skein algebras and cluster algebras of marked surfaces
• Groupoid-theoretical methods in the mapping class groups of surfaces
• A regular homotopy version of the Goldman-Turaev Lie bialgebra, the Enomoto-Satoh traces and the divergence cocycle in the Kashiwara-Vergne problem

Cited by in corpus (1)

• Classical shadows of stated skein representations at roots of unity