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Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths
About this Title
Sergey Fomin, Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 and Dylan Thurston, Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 255, Number 1223
ISBNs: 978-1-4704-2967-6 (print); 978-1-4704-4823-3 (online)
DOI: https://doi.org/10.1090/memo/1223
Published electronically: August 14, 2018
Keywords: Cluster algebra,
lambda length,
decorated Teichmüller space,
opened surface,
tagged triangulation,
shear coordinates,
integral lamination,
Ptolemy relations
MSC: Primary 13F60; Secondary 30F60, 57M50
Table of Contents
Chapters
- 1. Introduction
- 2. Non-normalized cluster algebras
- 3. Rescaling and normalization
- 4. Cluster algebras of geometric type and their positive realizations
- 5. Bordered surfaces, arc complexes, and tagged arcs
- 6. Structural results
- 7. Lambda lengths on bordered surfaces with punctures
- 8. Lambda lengths of tagged arcs
- 9. Opened surfaces
- 10. Lambda lengths on opened surfaces
- 11. Non-normalized exchange patterns from surfaces
- 12. Laminations and shear coordinates
- 13. Shear coordinates with respect to tagged triangulations
- 14. Tropical lambda lengths
- 15. Laminated Teichmüller spaces
- 16. Topological realizations of some coordinate rings
- 17. Principal and universal coefficients
- A. Tropical degeneration and relative lambda lengths
- B. Versions of Teichmüller spaces and coordinates
Abstract
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmüller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths.
Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths.
This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.
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