Mutation-periodic quivers, integrable maps and associated Poisson algebras

Area of Science:

• Mathematical Physics
• Integrable Systems
• Algebraic Geometry

Background:

• Introduces a class of maps derived from cluster mutation contexts.
• Reviews the foundational quiver context relevant to these mathematical structures.

Purpose of the Study:

• To analyze the Poisson bracket and algebra associated with specific functions linked to cluster mutation maps.
• To derive a bi-Hamiltonian structure for these maps.
• To demonstrate the complete integrability of the system and identify canonical coordinates.

Main Methods:

• Utilizes concepts from cluster algebras and quiver theory.
• Employs Poisson geometry to define brackets and algebras.
• Derives bi-Hamiltonian structures and canonical transformations.

Main Results:

• Establishes a bi-Hamiltonian structure for the maps.
• Constructs sequences of Poisson-commuting functions, proving complete integrability.
• Identifies the maps as canonical transformations and Bäcklund transformations related to Liouville's equation.

Conclusions:

• The derived bi-Hamiltonian structure facilitates the demonstration of complete integrability for cluster mutation maps.
• The maps serve as Bäcklund transformations, linking to fundamental equations like Liouville's equation.
• Canonical coordinates simplify the description of the map's behavior and its invariant functions.