Extensions in Jacobian algebras via punctured skein relations

TL;DR

The paper addresses classifying non-split extensions in Jacobian algebras arising from punctured surfaces, focusing on the punctured disk and type $D$ cluster categories. It develops a geometric framework using punctured skein relations to generate non-split triangles in the cluster category, and proves that all non-trivial triangles with indecomposable extremes in type $D$ arise from these skein relations. A dimension-preserving criterion $d(C)=d(igl\\{\alpha,\beta\bigl\})$ links multicurve skein terms to exact sequences in Jacobian algebras, and explicit cases for $e(eta,eta')=1$ and $2$ are worked out. The results yield a concrete, geometry-driven method to compute extensions across Jacobian algebras from punctured surfaces and extend naturally to other surfaces via cluster-tilting equivalences and reduction techniques, with detailed examples on the disk and beyond.

Abstract

Given a Jacobian algebra arising from the punctured disk, we show that all non-split extensions can be found using the tagged arcs and skein relations previously developed in cluster algebras theory. Our geometric interpretation can be used to find non-split extensions over other Jacobian algebras arising form surfaces with punctures. We show examples in type $D$ and in a punctured surface.

Figures (38)

• Figure 1: Non-split extension Lemma \ref{['new-sequences']}
• Figure 2: Possible middle terms for the extension. Lemma \ref{['middle-terms']}.
• Figure 3: Category of tagged arcs on the punctured disk. For each arc $\gamma$, $\tau(\gamma)$ is the arc immediate on the left.
• Figure 4: Tagged arcs at a puncture and noose. Self-folded triangle (right).
• Figure 5: Smoothing arcs.

Theorems & Definitions (28)

• Theorem
• Proposition
• Lemma 2.1
• proof
• Lemma 2.2
• Remark 2.3
• proof
• Lemma 2.4
• proof
• Definition 3.1