Cluster algebras and triangulated surfaces. Part II: Lambda lengths

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 Teichmueller 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.

99 pages, 46 figures; v2: new subsection "subsequent developments" and references [34]-[55]

References in corpus (7)

• The quantum dilogarithm and representations quantum cluster varieties
• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• A positive basis for surface skein algebras
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• On Quantizing Teichmüller and Thurston theories
• New Graphs of Finite Mutation Type
• Decorated Teichmüller Theory of Bordered Surfaces

Cited by in corpus (60)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• A positive basis for surface skein algebras
• Locally acyclic cluster algebras
• Cluster algebras and triangulated orbifolds
• Universal geometric cluster algebras
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• A geometric model for the derived category of gentle algebras
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras
• Skein algebras and cluster algebras of marked surfaces
• On genus expansion of superpolynomials
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• Line defects and (framed) BPS quivers
• A compendium on the cluster algebra and quiver package in sage
• Geometric realizations of the accordion complex of a dissection
• Bases for cluster algebras from orbifolds
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• Universal geometric coefficients for the once-punctured torus
• Combinatorial rigidity of arc complexes
• Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
• Combinatorial cluster expansion formulas from triangulated surfaces
• Cluster algebraic interpretation of infinite friezes
• Initial-seed recursions and dualities for d-vectors
• Exact WKB analysis and cluster algebras
• Complete gentle and special biserial algebras are $g$-tame
• Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings
• Bases for cluster algebras from orbifolds with one marked point
• About the Absence of Exotics and the Coulomb Branch Formula
• Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots
• Ginzburg algebras of triangulated surfaces and perverse schobers
• $F$-matrices of cluster algebras from triangulated surfaces
• Snake Graphs from Triangulated Orbifolds
• $\mathcal{G}$-systems
• Snake graph calculus and cluster algebras from surfaces II: Self-crossing snake graphs
• Universal geometric cluster algebras from surfaces
• Multitriangulations and tropical Pfaffians
• Positivity for quasi-cluster algebras
• A categorification of cluster algebras of type B and C through symmetric quivers
• Minimal Length Maximal Green Sequences and Triangulations of Polygons
• Quantisation Spaces of Cluster Algebras
• Cluster Expansions: T-walks, Labeled Posets and Matrix Calculations
• Laminations of punctured surfaces as $\tau$-regular irreducible components
• Dominance phenomena: mutation, scattering and cluster algebras
• An expansion formula for type A and Kronecker quantum cluster algebras
• Linear relations for Laurent polynomials and lattice equations
• Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients
• Universal geometric coefficients for the four-punctured sphere
• From frieze patterns to cluster categories
• Shellability and Sphericity of the quasi-arc complex of the Möbius strip
• Cluster expansion formulas and perfect matchings for type B and C
• On Structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces
• A multiplication formula for cluster characters in gentle algebras
• Ptolemy's equation and kin
• Extensions in Jacobian algebras via punctured skein relations
• Rooted Clusters for Graph LP Algebras
• Poisson and quantum geometry of acyclic cluster algebras
• Mutation of type $D$ friezes
• Denseness of $g$-vector cones from weighted orbifolds
• Cluster Algebras, Symplectic Leaves and Quantum Groups