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LINEAR RELATIONS AND INTEGRABILITY FOR CLUSTER ALGEBRAS FROM AFFINE QUIVERS

Part of: Sequences and sets Arithmetic rings and other special rings Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  13 August 2020

JOE PALLISTER
JOE PALLISTER*
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, UK, e-mail: [email protected]
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Abstract

We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

MSC classification

Primary: 13F60: Cluster algebras
Secondary: 37J35: Completely integrable systems, topological structure of phase space, integration methods 11B37: Recurrences
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 3 , September 2021 , pp. 584 - 621
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Assem, I., Reutenauer, C. and Smith, D., Friezes, Adv. Math. 225(6) (2010), 3134–3165.CrossRefGoogle Scholar
Caldero, P. and Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81(3) (2006), 595–616.CrossRefGoogle Scholar
Crawley-Boevey, W., Lectures on Representations of Quivers, Lectures in Oxford (1992).Google Scholar
Dodgson, C. L., IV. Condensation of determinants, being a new and brief method for computing their arithmetical values, Proc. Roy. Soc. Lond. 15 (1867), 150–155.CrossRefGoogle Scholar
Fomin, S., Williams, L. and Zelevinsky, A., Introduction to cluster algebras chapters 1–3, arXiv preprint arXiv:1608.05735 (2016).Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras i: foundations, J. Am. Math. Soc. 15(2) (2002), 497–529.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras ii: Finite type classification, Invent. Math. 154(1) (2003), 63–121.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras iv: coefficients, Compos. Math. 143(1) (2007), 112–164.CrossRefGoogle Scholar
Fordy, A. P., Mutation-periodic quivers, integrable maps and associated Poisson algebras, Philos. Trans. Roy. Soc. Lond. A Math. Phys. Eng. Sci. 369(1939) (2011), 1264–1279.Google ScholarPubMed
Fordy, A. P. and Hone, A., Discrete integrable systems and Poisson algebras from cluster maps, Commun. Math. Phys. 325(2) (2014), 527–584.CrossRefGoogle Scholar
Fordy, A. P. and Marsh, R. J., Cluster mutation-periodic quivers and associated Laurent sequences. J. Algebr. Comb. 34(1) (2011), 19–66.CrossRefGoogle Scholar
Gekhtman, M., Shapiro, M., Vainshtein, A., Cluster algebras and Weil-Petersson forms, Duke Math. J. 127(2) (2005), 291–311.CrossRefGoogle Scholar
Keller, B. and Scherotzke, S., Linear recurrence relations for cluster variables of affine quivers, Adv. Math. 228(3) (2011), 1842–1862.CrossRefGoogle Scholar
Maeda, S., Completely integrable symplectic mapping, Proc. Jpn. Acad. Ser. A, Math. Sci. 63(6) (1987), 198–200.CrossRefGoogle Scholar
Magri, F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19(5) (1978), 1156–1162.CrossRefGoogle Scholar
Veselov, A. P., Integrable maps, Russ. Math. Surv 46(5) (1991), 1.CrossRefGoogle Scholar