Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths
Sergey Fomin · Professor Dylan Thurston
Oct 2018 · American Mathematical Soc.
Ebook
98
Pages
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About this ebook
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 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.
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.
About the author
Sergey Fomin: University of Michigan, Ann Arbor, MI, USA,
Professor Dylan Thurston: Indiana University, Bloomington, IN, USA
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