Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

ISBN-13: 9781470429676 / Angielski

Sergey Fomin;Dylan Thurston

Opis

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 Teichmuller 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. 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.

Spis treści

• Introduction
• Non-normalized cluster algebras
• Rescaling and normalization
• Cluster algebras of geometric type and their positive realizations
• Bordered surfaces, arc complexes, and tagged arcs
• Structural results
• Lambda lengths on bordered surfaces with punctures
• Lambda lengths of tagged arcs
• Opened surfaces
• Lambda lengths on opened surfaces
• Non-normalized exchange patterns from surfaces
• Laminations and shear coordinates
• Shear coordinates with respect to tagged triangulations
• Tropical lambda lengths
• Laminated Teichmuller spaces
• Topological realizations of some coordinate rings
• Principal and universal coefficients
• Appendix A. Tropical degeneration and relative lambda lengths
• Appendix B. Versions of Teichmuller spaces and coordinates
• Bibliography