American Institute of Mathematical Sciences

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September  2014, 6(3): 297-318. doi: 10.3934/jgm.2014.6.297

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, Portugal
2. 

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

Received  July 2013 Revised  July 2014 Published  September 2014

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: cluster algebras, Symplectic reduction, Poisson maps, difference equations., symplectic maps.
Mathematics Subject Classification: Primary: 37J10, 53D20; Secondary: 13F60, 39A2.
Citation: Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297
References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin/Cummings Publishing Co. Inc., 1978.  Google Scholar

[2]

I. Cruz and M. E. Sousa-Dias, Reduction of order of cluster-type recurrence relations, São Paulo J. Math. Sci., 6 (2012), 203-225. Available from: http://www.ime.usp.br/~spjm//articlepdf/466.pdf. doi: 10.11606/issn.2316-9028.v6i2p203-225.  Google Scholar

[3]

J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-72923-7.  Google Scholar

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S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15 (2002), 497-529. doi: 10.1090/S0894-0347-01-00385-X.  Google Scholar

[5]

A. Fordy and A. Hone, Symplectic maps from cluster algebras, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 7 (2011), Paper 091, 12pp. doi: 10.3842/sigma.2011.091.  Google Scholar

[6]

A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys., 325 (2014), 527-584. doi: 10.1007/s00220-013-1867-y.  Google Scholar

[7]

A. Fordy and R. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66. doi: 10.1007/s10801-010-0262-4.  Google Scholar

[8]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, 167, AMS, Providence, RI, 2010. doi: 10.1090/surv/167.  Google Scholar

[9]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), 291-311. doi: 10.1215/S0012-7094-04-12723-X.  Google Scholar

[10]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), 899-934.  Google Scholar

[11]

A. Hone, Laurent polynomials and superintegrable maps, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 3 (2007), Paper 022, 18pp. doi: 10.3842/sigma.2007.022.  Google Scholar

[12]

A. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, preprint,, , ().   Google Scholar

[13]

A. Iatrou and J. Roberts, Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity, 15 (2002), 459-489. doi: 10.1088/0951-7715/15/2/313.  Google Scholar

[14]

B. Keller, Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories (eds. Thorsten Holm et al.), London Math. Soc. Lecture Note Ser., 375 (2010), 76-160. doi: 10.1017/cbo9781139107075.004.  Google Scholar

[15]

P. Libermann and C-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[16]

S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., NJ, 1964.  Google Scholar

[17]

G. Quispel, J. Roberts and C. Thompson, Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421. doi: 10.1016/0375-9601(88)90803-1.  Google Scholar

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin/Cummings Publishing Co. Inc., 1978.  Google Scholar

[2]

I. Cruz and M. E. Sousa-Dias, Reduction of order of cluster-type recurrence relations, São Paulo J. Math. Sci., 6 (2012), 203-225. Available from: http://www.ime.usp.br/~spjm//articlepdf/466.pdf. doi: 10.11606/issn.2316-9028.v6i2p203-225.  Google Scholar

[3]

J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-72923-7.  Google Scholar

[4]

S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15 (2002), 497-529. doi: 10.1090/S0894-0347-01-00385-X.  Google Scholar

[5]

A. Fordy and A. Hone, Symplectic maps from cluster algebras, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 7 (2011), Paper 091, 12pp. doi: 10.3842/sigma.2011.091.  Google Scholar

[6]

A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys., 325 (2014), 527-584. doi: 10.1007/s00220-013-1867-y.  Google Scholar

[7]

A. Fordy and R. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66. doi: 10.1007/s10801-010-0262-4.  Google Scholar

[8]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, 167, AMS, Providence, RI, 2010. doi: 10.1090/surv/167.  Google Scholar

[9]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), 291-311. doi: 10.1215/S0012-7094-04-12723-X.  Google Scholar

[10]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), 899-934.  Google Scholar

[11]

A. Hone, Laurent polynomials and superintegrable maps, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 3 (2007), Paper 022, 18pp. doi: 10.3842/sigma.2007.022.  Google Scholar

[12]

A. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, preprint,, , ().   Google Scholar

[13]

A. Iatrou and J. Roberts, Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity, 15 (2002), 459-489. doi: 10.1088/0951-7715/15/2/313.  Google Scholar

[14]

B. Keller, Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories (eds. Thorsten Holm et al.), London Math. Soc. Lecture Note Ser., 375 (2010), 76-160. doi: 10.1017/cbo9781139107075.004.  Google Scholar

[15]

P. Libermann and C-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[16]

S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., NJ, 1964.  Google Scholar

[17]

G. Quispel, J. Roberts and C. Thompson, Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421. doi: 10.1016/0375-9601(88)90803-1.  Google Scholar

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