Laurent phenomenon algebras arising from surfaces II: Laminated surfaces
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Laurent phenomenon algebras arising from surfaces II: Laminated surfaces Jon Wilson1
© Springer Nature Switzerland AG 2020
Abstract It was shown by Fock and Goncharov (Dual Teichmüller and lamination spaces. Handbook of Teichmüller Theory, 2007), and Fomin et al. (Acta Math 201(1):83–146, 2008) that some cluster algebras arise from triangulated orientable surfaces. Subsequently, Dupont and Palesi (J Algebraic Combinatorics 42(2):429–472, 2015) generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasicluster algebras. In Wilson (Int Math Res Notices 341, 2017) we linked this framework to Lam and Pylyavskyy’s Laurent phenomenon algebras (J Math 4(1):121–162, 2016), showing that unpunctured surfaces admit an LP structure. In this paper we extend quasi-cluster algebras to include punctured surfaces. Moreover, by adding laminations to the surface we demonstrate that all punctured and unpunctured surfaces admit LP structures. In short, we link two constructions which arose as seemingly unrelated generalisations of cluster algebras—one of the generalisations (quasi-cluster algebras) being based on triangulated surfaces, and the other (Laurent phenomenon algebras) based on the Laurent phenomenon. We thus provide a rich class of geometric examples in which to help study Laurent phenomenon algebras.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Laurent phenomenon algebras . . . . . . . . . . . 3 Quasi-cluster algebras . . . . . . . . . . . . . . . Quasi-seeds and mutation . . . . . . . . . . . . . 4 Anti-symmetric quivers . . . . . . . . . . . . . . 4.1 The double cover and anti-symmetric quivers
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The author gratefully acknowledges the support they received from the Teach@Tuebingen postdoctoral scheme and their EPSRC studentship.
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Jon Wilson [email protected] Department of Mathematics, Eberhard Karls University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany 0123456789().: V,-vol
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4.2 Mutation of anti-symmetric quivers via LP mutation . . . . . . . . 5 Laminated surfaces and their (laminated) quasi-cluster algebra. . . . . 5.1 Laminations and shear coordinates . . . . . . . . . . . . . . . . . 5.2 Opening the surface . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Transverse measure and tropical lambda lengths . . . . . . . . . . 5.4 Laminated lambda lengths and the laminated quasi-cluster algebra 6 Connecting laminated quasi-cluster algebras to LP algebras . . . . . . 6.1 Finding exchange relations of quasi-arcs via quivers . . . . . . . . 6.2 Laminated quasi-cluster algebras via LP mutation . . . . . . . . . 6.3 Principal lam
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