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Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths - Memoirs of the American Mathematical Society (Paperback)
  • Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths - Memoirs of the American Mathematical Society (Paperback)
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Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths - Memoirs of the American Mathematical Society (Paperback)

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Paperback 101 Pages / Published: 30/08/2018
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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 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.

Publisher: American Mathematical Society
ISBN: 9781470429676
Number of pages: 101
Dimensions: 254 x 178 mm

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