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We define a (co-)Poisson (co)algebra of curves on a bordered surface. Abordered surface is a surface whose boundary have marked points. Curves on thebordered surface are oriented loops and oriented arcs whose endpoints in theset of marked points. We define a (co-)Poisson (co)bracket on the symmetricalgebra of a quotient of the vector space spanned by the regular homotopyclasses of curves on the bordered surface by generalizing the Goldman bracketand the Turaev cobracket. Moreover, we define a Poisson algebra of unorientedcurves on a bordered surface and show that a quantization of the Poissonalgebra coincides with the skein algebra of the bordered surface defined byMuller.

Poisson algebras of curves on bordered surfaces and skein quantization