Softcover ISBN:  9781470429676 
Product Code:  MEMO/255/1223 
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Electronic ISBN:  9781470448233 
Product Code:  MEMO/255/1223.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 255; 2018; 98 ppMSC: Primary 13; Secondary 30; 57;
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors 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 nonnormalized cluster algebra and the machinery of tropical lambda lengths.
The authors' 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 the authors' previous paper, removing unnecessary assumptions on the surface. 
Table of Contents

Chapters

1. Introduction

2. Nonnormalized 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. Nonnormalized 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


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For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors 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 nonnormalized cluster algebra and the machinery of tropical lambda lengths.
The authors' 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 the authors' previous paper, removing unnecessary assumptions on the surface.

Chapters

1. Introduction

2. Nonnormalized 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. Nonnormalized 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