Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras

TL;DR

This paper explores the integrability of certain maps derived from cluster mutation, focusing on their Poisson structures, bi-Hamiltonian properties, and canonical coordinates, revealing their role as Bäcklund transformations.

Contribution

It introduces a bi-Hamiltonian structure for mutation-related maps, constructs commuting functions, and demonstrates their complete integrability and canonical form.

Findings

Established a Poisson algebra for the maps

Derived canonical coordinates and commuting functions

Showed the map acts as a Bäcklund transformation

Abstract

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville's equation and the map plays the role of a B\"acklund transformation.