Fordy, AP orcid.org/0000-0002-2523-0262 (2014) Periodic cluster mutations and related integrable maps. Journal of Physics A: Mathematical and Theoretical, 47 (47). 474003. ISSN 1751-8113
Abstract
One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the 'Laurent phenomenon'). There are many nonlinear recurrences which exhibit the Laurent phenomenon and thus unexpectedly generate integer sequences. The mutation of a typical quiver will not generate a recurrence, but rather an erratic sequence of exchange relations. How do we 'design' a quiver which gives rise to a given recurrence? A key role is played by the concept of 'periodic cluster mutation', introduced in 2009. Each recurrence corresponds to a finite dimensional map. In the context of cluster mutations, these are called 'cluster maps'. What properties do cluster maps have? Are they integrable in some standard sense? In this review I describe how integrable maps arise in the context of cluster mutations. I first explain the concept of 'periodic cluster mutation', giving some classification results. I then give a review of what is meant by an integrable map and apply this to cluster maps. Two classes of integrable maps are related to interesting monodromy problems, which generate interesting Poisson algebras of functions, used to prove complete integrability and a linearization. A connection to the Hirota–Miwa equation is explained.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | (c) 2014 IOP Publishing Ltd. This is an author produced version of a paper published in Journal of Physics A: Mathematical and Theoretical. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Poisson algebra; bi-Hamiltonian; integrable maps; super-integrability; Laurent property; cluster algebra |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 23 Feb 2018 13:28 |
Last Modified: | 21 Mar 2018 21:16 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1751-8113/47/47/474003 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:119302 |