We investigate two algebras consisting of curves on a surface with interior punctures -- the cluster algebra defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. We establish their compatibility, and use it to prove Roger-Yang's conjecture that the skein algebra is a deformation quantization of the decorated Teichmuller space. We also obtain several structural results on the cluster algebra of surfaces. The cluster algebra of a positive genus surface is not finitely generated, and it differs from its upper cluster algebra.

The authors would like to thank to Wade Bloomquist, Hyunkyu Kim, Thang Le, Kyungyong Lee, Gregg Musiker, Fan Qin, and Dylan Thurston for valuable conversations. This work was completed while the first author was visiting Stanford University. He gratefully appreciates the hospitality during his visit. The second author is partially supported by grant DMS-1906323 from the US National Science Foundation and a Birman Fellowship from the American Mathematical Society.