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Re-factorising a QRT map
1. | School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia |
2. | Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus |
A QRT map is the composition of two involutions on a biquadratic curve: one switching the $ x $-coordinates of two intersection points with a given horizontal line, and the other switching the $ y $-coordinates of two intersections with a vertical line. Given a QRT map, a natural question is to ask whether it allows a decomposition into further involutions. Here we provide new answers to this question and show how they lead to a new class of maps, as well as known HKY maps and quadrirational Yang-Baxter maps.
References:
[1] |
V. Adler,
Recutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.
doi: 10.1007/BF01085984. |
[2] |
V. Adler, A. Bobenko and Y. Suris,
Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings, Comm. Anal. Geom., 12 (2004), 967-1007.
doi: 10.4310/CAG.2004.v12.n5.a1. |
[3] |
J. Atkinson, Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups, preprint, arXiv: math/1301.4613. |
[4] |
J. Atkinson, P. Howes, N. Joshi and N. Nakazono, Geometry of an elliptic difference equation
related to Q4, J. Lond. Math. Soc. (2), 93 (2016), 763–784.
doi: 10.1112/jlms/jdw020. |
[5] |
J. Atkinson and M. Nieszporski,
Multi-quadratic quad equations: Integrable cases from a factorised-discriminant hypothesis, Int. Math. Res. Not. IMRN, 2014 (2013), 4215-4240.
doi: 10.1093/imrn/rnt066. |
[6] |
J. Atkinson and F. Nijhoff,
Solutions of Adler's lattice equation associated with 2-cycles of the Bäcklund transformation, J. Nonlinear Math. Phys., 15 (2008), 34-42.
doi: 10.2991/jnmp.2008.15.s3.4. |
[7] |
J. Atkinson and Y. Yamada, Quadrirational Yang-Baxter maps and the elliptic Cremona system, preprint, arXiv: math/1804.01794. |
[8] |
H. Baker, Principles of Geometry, Cambridge Library Collection, 1, Cambridge University
Press, Cambridge, 2010. |
[9] |
R. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London,
1982. |
[10] |
V. Bazhanov and S. Sergeev,
Yang-Baxter maps, discrete integrable equations and quantum groups, Nuclear Phys. B, 926 (2018), 509-543.
doi: 10.1016/j.nuclphysb.2017.11.017. |
[11] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Integrable Coxeter groups, Phys. Lett. A, 159 (1991), 221-232.
doi: 10.1016/0375-9601(91)90516-B. |
[12] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Rational mappings, arborescent iterations, and the symmetries of integrability, Phys. Rev. Lett., 67 (1991), 1373-1376.
doi: 10.1103/PhysRevLett.67.1373. |
[13] |
M. Bruschi, O. Ragnisco, P. Santini and T. Gui-Zhang,
Integrable symplectic maps, Phys. D, 49 (1991), 273-294.
doi: 10.1016/0167-2789(91)90149-4. |
[14] |
W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge,
1911. |
[15] |
V. Caudrelier, N. Crampé and Q. Zhang, Set-theoretical reflection equation: Classification of reflection maps, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/9/095203. |
[16] |
H. Coxeter,
Frieze patterns, Acta Arith., 18 (1971), 297-310.
doi: 10.4064/aa-18-1-297-310. |
[17] |
A. Dimakis and F. Müller-Hoissen,
Matrix Kadomtsev-Petviashvili equation: Tropical limit, Yang-Baxter and pentagon maps, Theoret. and Math. Phys., 196 (2018), 1164-1173.
doi: 10.1134/S0040577918080056. |
[18] |
A. Dimakis and F. Müller-Hoissen,
Matrix KP: Tropical limit and Yang-Baxter maps, Lett. Math. Phys., 109 (2019), 799-827.
doi: 10.1007/s11005-018-1127-3. |
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P. Etingof,
Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Commun. Algebra, 31 (2003), 1961-1973.
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P. Etingof, T. Schedler and A. Soloviev,
Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100 (1999), 169-209.
doi: 10.1215/S0012-7094-99-10007-X. |
[23] |
A. Fordy and P. Kassotakis,
Multidimensional maps of QRT type, J. Phys. A, 39 (2006), 10773-10786.
doi: 10.1088/0305-4470/39/34/012. |
[24] |
A. Fordy and P. Kassotakis, Integrable maps which preserve functions with symmetries, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/20/205201. |
[25] |
G. G. Grahovski, S. Konstantinou-Rizos and A. V. Mikhailov, Grassmann extensions of Yang-Baxter maps, J. Phys. A, 49 (2016), 17pp.
doi: 10.1088/1751-8113/49/14/145202. |
[26] |
G. Gubbiotti, N. Joshi, D. T. Tran and C.-M. Viallet, Complexity and integrability in 4D bi-rational maps with two invariants, preprint, arXiv: math/1808.04942. |
[27] |
F. Haggar, G. Byrnes, G. Quispel and H. Capel,
K-integrals and k-Lie symmetries in discrete dynamical systems, Phys. A, 233 (1996), 379-394.
doi: 10.1016/S0378-4371(96)00142-2. |
[28] |
J. Hietarinta,
Permutation-type solutions to the Yang-Baxter and other $n$-simplex equations, J. Phys. A, 30 (1997), 4757-4771.
doi: 10.1088/0305-4470/30/13/024. |
[29] |
J. Hietarinta and C. Viallet, On the parametrization of solutions of the Yang-Baxter equations, preprint, arXiv: math/9504028. |
[30] |
R. Hirota, K. Kimura and H. Yahagi,
How to find conserved quantities of nonlinear discrete equations, J. Phys. A, 34 (2001), 10377-10386.
doi: 10.1088/0305-4470/34/48/304. |
[31] |
A. Iatrou and J. Roberts,
Integrable mappings of the plane preserving biquadratic invariant curves, J. Phys. A, 34 (2001), 6617-6636.
doi: 10.1088/0305-4470/34/34/308. |
[32] |
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. |
[33] |
N. Iyudu and S. Shkarin et al., The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations, Duke Math. J., 164 (2015), 2539–2575.
doi: 10.1215/00127094-3146603. |
[34] |
M. Jimbo (ed.), Yang-Baxter Equation in Integrable Systems, Advanced Series in Mathematical Physics, 10, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
doi: 10.1142/1021. |
[35] |
J. Jogia, D. Roberts and F. Vivaldi,
An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149.
doi: 10.1088/0305-4470/39/5/008. |
[36] |
N. Joshi, B. Grammaticos, T. Tamizhmani and A. Ramani,
From integrable lattices to non-QRT mappings, Lett. Math. Phys., 78 (2006), 27-37.
doi: 10.1007/s11005-006-0103-5. |
[37] |
N. Joshi and C.-M. Viallet, Rational maps with invariant surfaces, J. Integrable Syst., 3 (2018), 14pp.
doi: 10.1093/integr/xyy017. |
[38] |
K. Kajiwara, M. Noumi and Y. Yamada,
Discrete dynamical systems with ${W}({A}_{m-1}^{(1)} \times {A}_{n-1}^{(1)})$ symmetry, Lett. Math. Phys., 60 (2002), 211-219.
doi: 10.1023/A:1016298925276. |
[39] |
P. Kassotakis, The Construction of Discrete Dynamical System, Ph.D thesis, University of Leeds, 2006. |
[40] |
P. Kassotakis, Invariants in separated variables: Yang-Baxter, entwining and transfer maps, SIGMA Symmetry Integrability Geom. Methods Appl., 15 (2019), 36pp.
doi: 10.3842/SIGMA.2019.048. |
[41] |
P. Kassotakis and N. Joshi,
Integrable non-QRT mappings of the plane, Lett. Math. Phys., 91 (2010), 71-81.
doi: 10.1007/s11005-009-0360-1. |
[42] |
P. Kassotakis and M. Nieszporski, Families of integrable equations, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 14pp.
doi: 10.3842/SIGMA.2011.100. |
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P. Kassotakis and M. Nieszporski,
On non-multiaffine consistent-around-the-cube lattice equations, Phys. Lett. A, 376 (2012), 3135-3140.
doi: 10.1016/j.physleta.2012.10.009. |
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P. Kassotakis and M. Nieszporski, $2^n-$rational maps, J. Phys. A, 50 (2017), 9pp.
doi: 10.1088/1751-8121/aa6dbd. |
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P. Kassotakis and M. Nieszporski, Difference systems in bond and face variables and non-potential versions of discrete integrable systems, J. Phys. A, 51 (2018), 21pp.
doi: 10.1088/1751-8121/aad4c4. |
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K. Kimura, H. Yahagi, R. Hirota, A. Ramani, B. Grammaticos and Y. Ohta,
A new class of integrable discrete systems, J. Phys. A, 35 (2002), 9205-9212.
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show all references
References:
[1] |
V. Adler,
Recutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.
doi: 10.1007/BF01085984. |
[2] |
V. Adler, A. Bobenko and Y. Suris,
Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings, Comm. Anal. Geom., 12 (2004), 967-1007.
doi: 10.4310/CAG.2004.v12.n5.a1. |
[3] |
J. Atkinson, Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups, preprint, arXiv: math/1301.4613. |
[4] |
J. Atkinson, P. Howes, N. Joshi and N. Nakazono, Geometry of an elliptic difference equation
related to Q4, J. Lond. Math. Soc. (2), 93 (2016), 763–784.
doi: 10.1112/jlms/jdw020. |
[5] |
J. Atkinson and M. Nieszporski,
Multi-quadratic quad equations: Integrable cases from a factorised-discriminant hypothesis, Int. Math. Res. Not. IMRN, 2014 (2013), 4215-4240.
doi: 10.1093/imrn/rnt066. |
[6] |
J. Atkinson and F. Nijhoff,
Solutions of Adler's lattice equation associated with 2-cycles of the Bäcklund transformation, J. Nonlinear Math. Phys., 15 (2008), 34-42.
doi: 10.2991/jnmp.2008.15.s3.4. |
[7] |
J. Atkinson and Y. Yamada, Quadrirational Yang-Baxter maps and the elliptic Cremona system, preprint, arXiv: math/1804.01794. |
[8] |
H. Baker, Principles of Geometry, Cambridge Library Collection, 1, Cambridge University
Press, Cambridge, 2010. |
[9] |
R. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London,
1982. |
[10] |
V. Bazhanov and S. Sergeev,
Yang-Baxter maps, discrete integrable equations and quantum groups, Nuclear Phys. B, 926 (2018), 509-543.
doi: 10.1016/j.nuclphysb.2017.11.017. |
[11] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Integrable Coxeter groups, Phys. Lett. A, 159 (1991), 221-232.
doi: 10.1016/0375-9601(91)90516-B. |
[12] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Rational mappings, arborescent iterations, and the symmetries of integrability, Phys. Rev. Lett., 67 (1991), 1373-1376.
doi: 10.1103/PhysRevLett.67.1373. |
[13] |
M. Bruschi, O. Ragnisco, P. Santini and T. Gui-Zhang,
Integrable symplectic maps, Phys. D, 49 (1991), 273-294.
doi: 10.1016/0167-2789(91)90149-4. |
[14] |
W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge,
1911. |
[15] |
V. Caudrelier, N. Crampé and Q. Zhang, Set-theoretical reflection equation: Classification of reflection maps, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/9/095203. |
[16] |
H. Coxeter,
Frieze patterns, Acta Arith., 18 (1971), 297-310.
doi: 10.4064/aa-18-1-297-310. |
[17] |
A. Dimakis and F. Müller-Hoissen,
Matrix Kadomtsev-Petviashvili equation: Tropical limit, Yang-Baxter and pentagon maps, Theoret. and Math. Phys., 196 (2018), 1164-1173.
doi: 10.1134/S0040577918080056. |
[18] |
A. Dimakis and F. Müller-Hoissen,
Matrix KP: Tropical limit and Yang-Baxter maps, Lett. Math. Phys., 109 (2019), 799-827.
doi: 10.1007/s11005-018-1127-3. |
[19] |
V. G. Drinfeld, On some unsolved problems in quantum group theory, in Quantum Groups, Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1–8.
doi: 10.1007/BFb0101175. |
[20] |
J. 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. |
[21] |
P. Etingof,
Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Commun. Algebra, 31 (2003), 1961-1973.
doi: 10.1081/AGB-120018516. |
[22] |
P. Etingof, T. Schedler and A. Soloviev,
Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100 (1999), 169-209.
doi: 10.1215/S0012-7094-99-10007-X. |
[23] |
A. Fordy and P. Kassotakis,
Multidimensional maps of QRT type, J. Phys. A, 39 (2006), 10773-10786.
doi: 10.1088/0305-4470/39/34/012. |
[24] |
A. Fordy and P. Kassotakis, Integrable maps which preserve functions with symmetries, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/20/205201. |
[25] |
G. G. Grahovski, S. Konstantinou-Rizos and A. V. Mikhailov, Grassmann extensions of Yang-Baxter maps, J. Phys. A, 49 (2016), 17pp.
doi: 10.1088/1751-8113/49/14/145202. |
[26] |
G. Gubbiotti, N. Joshi, D. T. Tran and C.-M. Viallet, Complexity and integrability in 4D bi-rational maps with two invariants, preprint, arXiv: math/1808.04942. |
[27] |
F. Haggar, G. Byrnes, G. Quispel and H. Capel,
K-integrals and k-Lie symmetries in discrete dynamical systems, Phys. A, 233 (1996), 379-394.
doi: 10.1016/S0378-4371(96)00142-2. |
[28] |
J. Hietarinta,
Permutation-type solutions to the Yang-Baxter and other $n$-simplex equations, J. Phys. A, 30 (1997), 4757-4771.
doi: 10.1088/0305-4470/30/13/024. |
[29] |
J. Hietarinta and C. Viallet, On the parametrization of solutions of the Yang-Baxter equations, preprint, arXiv: math/9504028. |
[30] |
R. Hirota, K. Kimura and H. Yahagi,
How to find conserved quantities of nonlinear discrete equations, J. Phys. A, 34 (2001), 10377-10386.
doi: 10.1088/0305-4470/34/48/304. |
[31] |
A. Iatrou and J. Roberts,
Integrable mappings of the plane preserving biquadratic invariant curves, J. Phys. A, 34 (2001), 6617-6636.
doi: 10.1088/0305-4470/34/34/308. |
[32] |
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. |
[33] |
N. Iyudu and S. Shkarin et al., The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations, Duke Math. J., 164 (2015), 2539–2575.
doi: 10.1215/00127094-3146603. |
[34] |
M. Jimbo (ed.), Yang-Baxter Equation in Integrable Systems, Advanced Series in Mathematical Physics, 10, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
doi: 10.1142/1021. |
[35] |
J. Jogia, D. Roberts and F. Vivaldi,
An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149.
doi: 10.1088/0305-4470/39/5/008. |
[36] |
N. Joshi, B. Grammaticos, T. Tamizhmani and A. Ramani,
From integrable lattices to non-QRT mappings, Lett. Math. Phys., 78 (2006), 27-37.
doi: 10.1007/s11005-006-0103-5. |
[37] |
N. Joshi and C.-M. Viallet, Rational maps with invariant surfaces, J. Integrable Syst., 3 (2018), 14pp.
doi: 10.1093/integr/xyy017. |
[38] |
K. Kajiwara, M. Noumi and Y. Yamada,
Discrete dynamical systems with ${W}({A}_{m-1}^{(1)} \times {A}_{n-1}^{(1)})$ symmetry, Lett. Math. Phys., 60 (2002), 211-219.
doi: 10.1023/A:1016298925276. |
[39] |
P. Kassotakis, The Construction of Discrete Dynamical System, Ph.D thesis, University of Leeds, 2006. |
[40] |
P. Kassotakis, Invariants in separated variables: Yang-Baxter, entwining and transfer maps, SIGMA Symmetry Integrability Geom. Methods Appl., 15 (2019), 36pp.
doi: 10.3842/SIGMA.2019.048. |
[41] |
P. Kassotakis and N. Joshi,
Integrable non-QRT mappings of the plane, Lett. Math. Phys., 91 (2010), 71-81.
doi: 10.1007/s11005-009-0360-1. |
[42] |
P. Kassotakis and M. Nieszporski, Families of integrable equations, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 14pp.
doi: 10.3842/SIGMA.2011.100. |
[43] |
P. Kassotakis and M. Nieszporski,
On non-multiaffine consistent-around-the-cube lattice equations, Phys. Lett. A, 376 (2012), 3135-3140.
doi: 10.1016/j.physleta.2012.10.009. |
[44] |
P. Kassotakis and M. Nieszporski, $2^n-$rational maps, J. Phys. A, 50 (2017), 9pp.
doi: 10.1088/1751-8121/aa6dbd. |
[45] |
P. Kassotakis and M. Nieszporski, Difference systems in bond and face variables and non-potential versions of discrete integrable systems, J. Phys. A, 51 (2018), 21pp.
doi: 10.1088/1751-8121/aad4c4. |
[46] |
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