Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 048, 36 pages      arXiv:1901.01609

Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Pavlos Kassotakis
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

Received January 16, 2019, in final form June 15, 2019; Published online June 25, 2019

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole $F$ and $H$-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the $H$-list of Yang-Baxter maps can be considered as the $(k-1)$-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to $k$-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.

Key words: discrete integrable systems; Yang-Baxter maps; entwining maps; transfer maps.

pdf (636 kb)   tex (41 kb)  


  1. Adler V.E., Recuttings of polygons, Funct. Anal. Appl. 27 (1993), 141-143.
  2. Adler V.E., Bäcklund transformation for the Krichever-Novikov equation, Int. Math. Res. Not. 1998 (1998), 1-4, arXiv:solv-int/9707015.
  3. Adler V.E., On a class of third order mappings with two rational invariants, arXiv:nlin.SI/0606056.
  4. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, arXiv:nlin.SI/0202024.
  5. Adler V.E., Bobenko A.I., Suris Yu.B., Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings, Comm. Anal. Geom. 12 (2004), 967-1007, arXiv:math.QA/0307009.
  6. Adler V.E., Shabat A.B., Dressing chain for the acoustic spectral problem, Theoret. and Math. Phys. 149 (2006), 1324-1337, arXiv:nlin.SI/0604008.
  7. Adler V.E., Yamilov R.I., Explicit auto-transformations of integrable chains, J. Phys. A: Math. Gen. 27 (1994), 477-492.
  8. Atkinson J., Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups, arXiv:1301.4613.
  9. Atkinson J., Nieszporski M., Multi-quadratic quad equations: integrable cases from a factorized-discriminant hypothesis, Int. Math. Res. Not. 2014 (2014), 4215-4240, arXiv:1204.0638.
  10. Atkinson J., Yamada Y., Quadrirational Yang-Baxter maps and the elliptic Cremona system, arXiv:1804.01794.
  11. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  12. Bazhanov V.V., Sergeev S.M., Yang-Baxter maps, discrete integrable equations and quantum groups, Nuclear Phys. B 926 (2018), 509-543, arXiv:1501.06984.
  13. Boll R., Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 18 (2011), 337-365, arXiv:1009.4007.
  14. Bruschi M., Ragnisco O., Santini P.M., Tu G.Z., Integrable symplectic maps, Phys. D 49 (1991), 273-294.
  15. Capel H.W., Sahadevan R., A new family of four-dimensional symplectic and integrable mappings, Phys. A 289 (2001), 86-106.
  16. Cresswell C., Joshi N., The discrete first, second and thirty-fourth Painlevé hierarchies, J. Phys. A: Math. Gen. 32 (1999), 655-669.
  17. Dimakis A., Müller-Hoissen F., Simplex and polygon equations, SIGMA 11 (2015), 042, 49 pages, arXiv:1409.7855.
  18. Dimakis A., Müller-Hoissen F., Matrix Kadomtsev-Petviashvili equation: tropical limit, Yang-Baxter and pentagon maps, Theoret. and Math. Phys. 196 (2018), 1164-1173, arXiv:1709.09848.
  19. Dimakis A., Müller-Hoissen F., Matrix KP: tropical limit and Yang-Baxter maps, Lett. Math. Phys. 109 (2019), 799-827, arXiv:1708.05694.
  20. Doliwa A., Non-commutative rational Yang-Baxter maps, Lett. Math. Phys. 104 (2014), 299-309, arXiv:1308.2824.
  21. Drinfel'd V.G., On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 1-8.
  22. Duistermaat J.J., Discrete integrable systems. QRT maps and elliptic surfaces, Springer Monographs in Mathematics, Springer, New York, 2010.
  23. Etingof P., Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Comm. Algebra 31 (2003), 1961-1973, arXiv:math.QA/0112278.
  24. Etingof P., Schedler T., Soloviev A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209, arXiv:math.QA/9801047.
  25. Evripidou C.A., Kassotakis P., Vanhaecke P., Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems, Regul. Chaotic Dyn. 22 (2017), 721-739, arXiv:1709.06763.
  26. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, arXiv:math.RT/0104151.
  27. Fomin S., Zelevinsky A., The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), 119-144, arXiv:math.CO/0104241.
  28. Fordy A.P., Hone A., Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys. 325 (2014), 527-584, arXiv:1207.6072.
  29. Fordy A.P., Kassotakis P., Multidimensional maps of QRT type, J. Phys. A: Math. Gen. 39 (2006), 10773-10786.
  30. Fordy A.P., Kassotakis P., Integrable maps which preserve functions with symmetries, J. Phys. A: Math. Theor. 46 (2013), 205201, 12 pages, arXiv:1301.1927.
  31. Grahovski G.G., Konstantinou-Rizos S., Mikhailov A.V., Grassmann extensions of Yang-Baxter maps, J. Phys. A: Math. Theor. 49 (2016), 145202, 17 pages, arXiv:1510.06913.
  32. Hay M., Hierarchies of nonlinear integrable $q$-difference equations from series of Lax pairs, J. Phys. A: Math. Theor. 40 (2007), 10457-10471.
  33. Hietarinta J., Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax, J. Nonlinear Math. Phys. 26 (2019), 358-389, arXiv:1806.08511.
  34. Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016.
  35. Iatrou A., Higher dimensional integrable mappings, Phys. D 179 (2003), 229-253.
  36. Jimbo M. (Editor), Yang-Baxter equation in integrable systems, Adv. Ser. Math. Phys., Vol. 10, World Sci. Publ., Teaneck, NJ, 1989.
  37. Joshi N., Kassotakis P., Re-factorising a QRT map, arXiv:1906.00501.
  38. Joshi N., Nakazono N., Shi Y., Lattice equations arising from discrete Painlevé systems. I. $(A_2 + A_1)^{(1)}$ and $\big(A_1 + A_1^\prime\big)^{(1)}$ cases, J. Math. Phys. 56 (2015), 092705, 25 pages, arXiv:1401.7044.
  39. Joshi N., Nakazono N., Shi Y., Lattice equations arising from discrete Painlevé systems: II. $A^{(1)}_4$ case, J. Phys. A: Math. Theor. 49 (2016), 495201, 39 pages, arXiv:1603.09414.
  40. Kajiwara K., Noumi M., Yamada Y., Discrete dynamical systems with $W\big(A_{m-1}^{(1)}\times A_{n-1}^{(1)}\big)$ symmetry, Lett. Math. Phys. 60 (2002), 211-219, arXiv:nlin.SI/0106029.
  41. Kashaev R.M., On discrete three-dimensional equations associated with the local Yang-Baxter relation, Lett. Math. Phys. 38 (1996), 389-397, arXiv:solv-int/9512005.
  42. Kashaev R.M., Korepanov I.G., Sergeev S.M., Functional tetrahedron equation, Theoret. and Math. Phys. 117 (1998), 1402-1413, arXiv:solv-int/9801015.
  43. Kassotakis P., The construction of discrete dynamical system, Ph.D. Thesis, University of Leeds, 2006.
  44. Kassotakis P., Nieszporski M., Families of integrable equations, SIGMA 7 (2011), 100, 14 pages, arXiv:1106.0636.
  45. Kassotakis P., Nieszporski M., On non-multiaffine consistent around the cube lattice equations, Phys. Lett. A 376 (2012), 3135-3140, arXiv:1106.0435.
  46. Kassotakis P., Nieszporski M., $2^n$-rational maps, J. Phys. A: Math. Theor. 50 (2017), 21LT01, 9 pages, arXiv:1512.00771.
  47. Kassotakis P., Nieszporski M., Difference systems in bond and face variables and non-potential versions of discrete integrable systems, J. Phys. A: Math. Theor. 51 (2018), 385203, 21 pages, arXiv:1710.11111.
  48. Konstantinou-Rizos S., Mikhailov A.V., Darboux transformations, finite reduction groups and related Yang-Baxter maps, J. Phys. A: Math. Theor. 46 (2013), 425201, 16 pages.
  49. Korepanov I.G., Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics, arXiv:solv-int/9506003.
  50. Kouloukas T.E., Relativistic collisions as Yang-Baxter maps, Phys. Lett. A 381 (2017), 3445-3449, arXiv:1706.06361.
  51. Kouloukas T.E., Papageorgiou V.G., Entwining Yang-Baxter maps and integrable lattices, in Algebra, Geometry and Mathematical Physics,Banach Center Publ., Vol. 93, Polish Acad. Sci. Inst. Math., Warsaw, 2011, 163-175.
  52. Maeda S., Completely integrable symplectic mapping, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 198-200.
  53. Maillet J.-M., Nijhoff F., Integrability for multidimensional lattice models, Phys. Lett. B 224 (1989), 389-396.
  54. Maillet J.-M., Nijhoff F., The tetrahedron equation and the four-simplex equation, Phys. Lett. A 134 (1989), 221-228.
  55. McLachlan R.I., Quispel G.R.W., Generating functions for dynamical systems with symmetries, integrals, and differential invariants, Phys. D 112 (1998), 298-309.
  56. Nieszporski M., Kassotakis P., Systems of difference equations on a vector valued function that admit a 3D vector space of scalar potentials, in preparation.
  57. Noumi M., Yamada Y., Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281-295, arXiv:math.QA/9804132.
  58. Papageorgiou V.G., Nijhoff F.W., Capel H.W., Integrable mappings and nonlinear integrable lattice equations, Phys. Lett. A 147 (1990), 106-114.
  59. Papageorgiou V.G., Suris Yu.B., Tongas A.G., Veselov A.P., On quadrirational Yang-Baxter maps, SIGMA 6 (2010), 033, 9 pages, arXiv:0911.2895.
  60. Papageorgiou V.G., Tongas A.G., Veselov A.P., Yang-Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys. 47 (2006), 083502, 16 pages, arXiv:math.QA/0605206.
  61. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations, Phys. Lett. A 126 (1988), 419-421.
  62. Roberts J.A.G., Quispel G.R.W., Creating and relating three-dimensional integrable maps, J. Phys. A: Math. Gen. 39 (2006), L605-L615.
  63. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  64. Sergeev S.M., Solutions of the functional tetrahedron equation connected with the local Yang-Baxter equation for the ferro-electric condition, Lett. Math. Phys. 45 (1998), 113-119, arXiv:solv-int/9709006.
  65. Sklyanin E.K., Classical limits of ${\rm SU}(2)$-invariant solutions of the Yang-Baxter equation, J. Sov. Math. 40 (1988), 93-107.
  66. Suris Yu.B., Veselov A.P., Lax matrices for Yang-Baxter maps, J. Nonlinear Math. Phys. 10 (2003), suppl. 2, 223-230, arXiv:math.QA/0304122.
  67. Tsuda T., Integrable mappings via rational elliptic surfaces, J. Phys. A: Math. Gen. 37 (2004), 2721-2730.
  68. Veselov A.P., Integrable maps, Russian Math. Surveys 46 (1991), no. 5, 1-51.
  69. Veselov A.P., Yang-Baxter maps and integrable dynamics, Phys. Lett. A 314 (2003), 214-221, arXiv:math.QA/0205335.
  70. Veselov A.P., Yang-Baxter maps: dynamical point of view, in Combinatorial Aspect of Integrable Systems, MSJ Mem., Vol. 17, Math. Soc. Japan, Tokyo, 2007, 145-167.
  71. Veselov A.P., Shabat A.B., Dressing chains and the spectral theory of the Schrödinger operatorr, Funct. Anal. Appl. 27 (1993), 81-96.
  72. Viallet C.M., Integrable lattice maps: $Q_{\rm V}$, a rational version of $Q_4$, Glasg. Math. J. 51 (2009), 157-163, arXiv:0802.0294.
  73. Yang C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.

Previous article  Next article  Contents of Volume 15 (2019)