The Journal of Geometric Mechanics (JGM)

Reduction of cluster iteration maps

Pages: 297 - 318, Volume 6, Issue 3, September 2014      doi:10.3934/jgm.2014.6.297

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Inês Cruz - Centro de Matemática da Universidade do Porto (CMUP), Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, R. Campo Alegre, 687, 4169-007 Porto, Portugal (email)
M. Esmeralda Sousa-Dias - Center for Mathematical Analysis, Geometry and Dynamical Systems (CAMGSD), Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (email)

Abstract: We study iteration maps of difference equations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and presymplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several periodic quivers using either the presymplectic reduction or a Poisson reduction via log-canonical Poisson structures.

Keywords:  Symplectic reduction, cluster algebras, symplectic maps, Poisson maps, difference equations.
Mathematics Subject Classification:  Primary: 37J10, 53D20; Secondary: 13F60, 39A20.

Received: July 2013;      Revised: July 2014;      Available Online: September 2014.