Consequences of the compatibility of skein algebra and cluster algebra on surfaces

Abstract

We investigate two algebra of curves on a topological surface with punctures - the cluster algebra of surfaces defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. By establishing their compatibility, we resolve Roger-Yang's conjecture on the 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.

Figures (14)

• Figure 3.1: Three puzzle pieces and their associated matrix minors
• Figure 3.2: The fourth puzzle pieces and their associated matrix minors
• Figure 3.3: On the left, the envelope $\gamma_{v}^{w}$ encircles its jewel $v$. On the right is the corresponding dangle $d_{v}^{w}$, with the taggings necessarily distinct at $v$. In this example, both the tags are plain at $w$, but both could be notched at $w$ instead.
• Figure 3.4: The four tagged puzzle pieces. They are the images under $\tau$ of the four ordinary puzzle pieces from Figures \ref{['fig:puzzle']} and \ref{['fig:puzzle4']}.
• Figure 4.1: In Case 1, two type A puzzle pieces are glued along exactly one edge $\alpha$. The induced exchange relation from flipping $\alpha$ is $\alpha \alpha' = e_{1}e_{3} + e_{2}e_{4}$

Theorems & Definitions (73)

• Theorem A
• Theorem B
• Remark 1.1
• Theorem C
• Theorem D
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Example 2.4
• Lemma 2.5