The Roger-Yang skein algebra and the decorated Teichmuller space

TL;DR

This paper studies the Roger-Yang skein algebra related to decorated Teichmuller space, proving injectivity of a key homomorphism and absence of zero divisors for certain punctured surfaces, with implications for hyperbolic geometry.

Contribution

It establishes the injectivity of the Poisson algebra homomorphism and the lack of zero divisors in the skein algebra for a class of punctured surfaces, extending previous geometric algebra results.

Findings

Poisson algebra homomorphism is injective for specified surfaces.

Skein algebra has no zero divisors for these surfaces.

Generalized corner coordinates for normal arcs are introduced.

Abstract

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmuller space. In this paper, we consider surfaces with punctures which is not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang's Poisson algebra homomorphism is injective, and the skein algebra they defined have no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.