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Infinite rank surface cluster algebras

Lookup NU author(s): Dr Ilke CanakciORCiD

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Abstract

© 2019 Elsevier Inc. We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation sequences. We show transitivity of infinite mutation sequences on triangulations of an infinite surface and examine different types of mutation sequences. Moreover, we use a hyperbolic structure on an infinite surface to extend the notion of surface cluster algebras to infinite rank by giving cluster variables as lambda lengths of arcs. Furthermore, we study the structural properties of infinite rank surface cluster algebras in combinatorial terms, namely we extend “snake graph combinatorics” to give an expansion formula for cluster variables. We also show skein relations for infinite rank surface cluster algebras.

Publication metadata

Author(s): Çanakçı I, Felikson A

Publication type: Article

Publication status: Published

Journal: Advances in Mathematics

Year: 2019

Volume: 352

Pages: 862-942

Print publication date: 20/08/2019

Online publication date: 26/06/2019

Acceptance date: 20/05/2019

ISSN (print): 0001-8708

ISSN (electronic): 1090-2082

Publisher: Academic Press Inc.

URL: https://doi.org/10.1016/j.aim.2019.06.008

DOI: 10.1016/j.aim.2019.06.008

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