Reduction of cluster iteration maps

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), 899-934.
[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.
[12]	A. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, preprint, arXiv:1405.5379.
[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.
[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.
[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.
[16]	S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., NJ, 1964.
[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.

show all references

References:

[1]	R. Abraham and J. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin/Cummings Publishing Co. Inc., 1978.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), 899-934.
[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.
[12]	A. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, preprint, arXiv:1405.5379.
[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.
[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.
[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.
[16]	S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., NJ, 1964.
[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.

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