Reduction of cluster iteration maps

Inês Cruz; M. Esmeralda Sousa-Dias

doi:10.3934/jgm.2014.6.297

Article Contents

2014, Volume 6, Issue 3: 297-318. Doi: 10.3934/jgm.2014.6.297

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Reduction of cluster iteration maps

Inês Cruz^1, and
M. Esmeralda Sousa-Dias^2,

1.
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
2.
Center for Mathematical Analysis, Geometry and Dynamical Systems (CAMGSD), Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa

Received: June 30, 2013

Revised: June 30, 2014

Published: August 2014

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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.

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Mathematics Subject Classification: Primary: 37J10, 53D20; Secondary: 13F60, 39A20.

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