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Deep Points of Cluster Algebras

James Beyer, Greg Muller

Abstract

We initiate a systematic study of the deep points of a cluster algebra; that is, the points in the associated variety which are not in any cluster torus. We describe the deep points of cluster algebras of type A, rank 2, Markov, and unpunctured surface type.

Deep Points of Cluster Algebras

Abstract

We initiate a systematic study of the deep points of a cluster algebra; that is, the points in the associated variety which are not in any cluster torus. We describe the deep points of cluster algebras of type A, rank 2, Markov, and unpunctured surface type.
Paper Structure (42 sections, 51 theorems, 91 equations, 18 figures)

This paper contains 42 sections, 51 theorems, 91 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Cluster varieties and cluster tori
  3. Deep points
  4. Deep points and defects
  5. Summary of results
  6. Related work and acknowledgements
  7. Deep points in general cluster varieties
  8. Cluster varieties
  9. Deep points
  10. Deep points and upper cluster algebras
  11. Deep points of cluster algebras of polygons
  12. The cluster algebra(s) of a polygon
  13. Vanishing lemmas
  14. Deep points
  15. Deep points of cluster algebras of rank 2
  16. ...and 27 more sections

Key Result

Theorem \ref{thm: properties no deep points}

Let $\mathcal{A}$ be a cluster algebra with no deep points for all fields $\Bbbk$. Then

Figures (18)

  • Figure 1: A quadralateral with Ptolemy relation $EF=AC+BD$
  • Figure 2: The unique deep point in the $A_3$ cluster variety (with no frozens)
  • Figure 3: The Ptolemy relations
  • Figure 4: Modifying a triangulation to reduce the number of vanishing arcs.
  • Figure 5: A general deep point in the cluster algebra of a hexagon
  • ...and 13 more figures

Theorems & Definitions (126)

  • proof : Example 1.1
  • proof : Example 1.2
  • Theorem \ref{thm: properties no deep points}
  • proof : Warning 1.3
  • proof : Remark 2.1
  • proof : Warning 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 116 more