Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

By Sergey Fomin, Dylan Thurston. Memoirs of the American Mathematical Society, 2012

130 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bc4b860a87bdd3c7318d57396e2456cf57092dd2

Citations in the corpus

• A geometric model for the derived category of gentle algebras (2018, unknown): “ that ˜γ1 and ˜γ2 intersect at ˜q. • If qis a puncture, then we will choose “lifts” of γ1 and γ2 in the following way. Let ζbe a circle around the puncture (one can think of ζas of a horocycle, as in [40]). Assume that ζintersects with γ1 (and γ2) exactly once. By forgetting the part of γ1 and γ2 be”