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LINEAR RELATIONS AND INTEGRABILITY FOR CLUSTER ALGEBRAS FROM AFFINE QUIVERS

Part of: Arithmetic rings and other special rings Sequences and sets 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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