×

\(2^n\)-rational maps. (English) Zbl 1367.81088

Summary: We present a natural extension of the notion of nondegenerate rational maps (quadrirational maps) to arbitrary dimensions. We refer to these maps as \(2^n\)-rational maps. In this note we construct a rich family of \(2^n\)-rational maps. These maps by construction are involutions and highly symmetric in the sense that the maps and their companion maps have the same functional form.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T25 Yang-Baxter equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Etingof P 2003 Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation Commun. Algebra31 1961-73 · Zbl 1020.17008 · doi:10.1081/AGB-120018516
[2] Sklyanin E K 1988 Classical limits of SU(2)-invariant solutions of the Yang-Baxter equation J. Soviet Math.40 93-107 · Zbl 0636.58041 · doi:10.1007/BF01084941
[3] Drinfeld V G 1992 On some unsolved problems in quantum group theory Quantum Groups(Lecture Notes in Mathematics vol 1510) (New York: Springer) pp 1-8 · Zbl 0765.17014 · doi:10.1007/BFb0101175
[4] Yang C N 1967 Some exact results for the many body problem in one-dimension repulsive delta-function interaction Phys. Rev. Lett.19 1312-5 · Zbl 0152.46301 · doi:10.1103/PhysRevLett.19.1312
[5] Baxter R J 1972 Partition function for the eight-vertex lattice model Ann. Phys.70 193-22 · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1
[6] Berenstein A and Kazhdan D 1992 Geometric and unipotent crystals Geom. Funct. Anal.Special vol part I 188-236 (arXiv:math/9912105) · Zbl 1044.17006
[7] Veselov A P 2003 Yang-Baxter maps and integrable dynamics Phys. Lett. A 314 214-21 · Zbl 1051.81014 · doi:10.1016/S0375-9601(03)00915-0
[8] Adler V E, Bobenko A I and Suris Yu B 2004 Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings Commun. Anal. Geom.12 967-1007 · Zbl 1065.14015 · doi:10.4310/CAG.2004.v12.n5.a1
[9] Noumi M and Yamada Y 1998 Affine weyl groups, discrete dynamical systems and Painlevé equations Commun. Math. Phys.199 281-95 · Zbl 0952.37031 · doi:10.1007/s002200050502
[10] Kajiwara K, Noumi M and Yamada Y 2002 Discrete dynamical systems with w(am−1(1)×an−1(1)) symmetry Lett. Math. Phys.60 211-9 · Zbl 1077.37046 · doi:10.1023/A:1016298925276
[11] Papageorgiou V G, Tongas A G and Veselov A P 2006 Yang-Baxter maps and symmetries of integrable equations on quad-graphs J. Math. Phys.47 083502 · Zbl 1112.82017 · doi:10.1063/1.2227641
[12] Papageorgiou V G, Suris Yu B, Tongas A G and Veselov A P 2010 On quadrirational Yang-Baxter maps SIGMA6 033 · Zbl 1190.14012 · doi:10.3842/SIGMA.2010.033
[13] Kassotakis P and Nieszporski M 2012 On non-multiaffine consistent around the cube lattice equations Phys. Lett. A 376 3135-40 · Zbl 1266.37038 · doi:10.1016/j.physleta.2012.10.009
[14] Kassotakis P and Nieszporski M 2011 Families of integrable equations SIGMA100 14 · Zbl 1242.82010 · doi:10.3842/SIGMA.2011.100
[15] Kassotakis P and Nieszporski M 2017 Systems of difference equations on a vector valued function that admit 3d vector space of scalar potentials (in preparation)
[16] Atkinson J 2012 A multidimensionally consistent version of Hirota’s discrete KdV equation J. Phys. A: Math. Theor.45 222001 · Zbl 1250.35155 · doi:10.1088/1751-8113/45/22/222001
[17] Atkinson J 2013 Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups (arXiv:1301.4613 [nlin.SI])
[18] Atkinson J and Yamada Y 2014 Quadrirational Yang-Baxter maps and the E 8 Painlevé lattice (arXiv:1405.2745 [nlin.SI])
[19] Konstantinou-Rizos S and Mikhailov A V 2013 Darboux transformations, finite reduction groups and related Yang-Baxter maps J. Phys. A: Math. Theor.46 425201 · Zbl 1276.81069 · doi:10.1088/1751-8113/46/42/425201
[20] Kouloukas T E and Papageorgiou V G 2011 Entwining Yang-Baxter maps and integrable lattices Banach Center Publ.93 163-75 · Zbl 1248.81087 · doi:10.4064/bc93-0-13
[21] Bazhanov V V and Sergeev S M 2016 Yang-Baxter maps, discrete integrable equations and quantum groups (arXiv:1501.06984v2 [math-ph])
[22] Quispel G R W, Roberts J A G and Thompson C J 1988 Integrable mappings and soliton equations Phys. Lett. A 126 419-21 · Zbl 0679.58023 · doi:10.1016/0375-9601(88)90803-1
[23] Maeda S 1987 Completely integrable symplectic mapping Proc. Japan Acad. A 63 198-200 · Zbl 0634.58006 · doi:10.3792/pjaa.63.198
[24] Veselov A P 1991 Integrable maps Russ. Math. Surv.46 1-51 · Zbl 0785.58027 · doi:10.1070/RM1991v046n05ABEH002856
[25] Bruschi M, Ragnisco O, Santini P M and Gui-Zhang T 1991 Integrable symplectic maps Physica D 49 273-94 · Zbl 0734.58023 · doi:10.1016/0167-2789(91)90149-4
[26] Tsuda T 2004 Integrable mappings via rational elliptic surfaces J. Phys. A: Math. Gen.37 2721-30 · Zbl 1060.14051 · doi:10.1088/0305-4470/37/7/014
[27] Joshi N and Kassotakis P 2017 On a re-factorisation of the QRT map (in preparation)
[28] Kassotakis P and Joshi N 2010 Integrable non-QRT mappings of the plane Lett. Math. Phys.91 71-81 · Zbl 1187.39008 · doi:10.1007/s11005-009-0360-1
[29] Capel H W and Sahadevan R 2001 A new family of four-dimensional symplectic and integrable mappings Physica A 289 86-106 · Zbl 0971.37511 · doi:10.1016/S0378-4371(00)00314-9
[30] Iatrou A 2003 Higher dimensional integrable mappings Physica D 179 229-53 · Zbl 1020.37037 · doi:10.1016/S0167-2789(03)00011-3
[31] Roberts J A G and Quispel G R W 2006 Creating and relating three-dimensional integrable maps J. Phys. A: Math. Gen.39 L605-15 · Zbl 1370.37112 · doi:10.1088/0305-4470/39/42/L03
[32] Fordy A P and Kassotakis P G 2006 Multidimensional maps of QRT type J. Phys. A: Math. Gen.39 10773-86 · Zbl 1116.37037 · doi:10.1088/0305-4470/39/34/012
[33] Adler V E 2006 On a class of third order mappings with two rational invariants (arXiv:nlin/0606056)
[34] Korepanov I G 1995 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)
[35] Kashaev R M, Korepanov I G and Sergeev S M 1998 Functional tetrahedron equation Theor. Math. Phys.117 1402-13 · Zbl 0941.82018 · doi:10.1007/BF02557179
[36] Sergeev S M 1998 Solutions of the functional tetrahedron equation connected with the local Yang-Baxter equation for the ferro-electric condition Lett. Math. Phys.45 113-9 · Zbl 0908.35111 · doi:10.1023/A:1007483621814
[37] Maillet J M and Nijhoff F 1989 The tetrahedron equation and the four-simplex equation Phys. Lett. A 134 221-8 · doi:10.1016/0375-9601(89)90400-3
[38] Maillet J M and Nijhoff F 1989 Integrability for multidimensional lattice models Phys. Lett. B 224 389-96 · doi:10.1016/0370-2693(89)91466-4
[39] Dimakis A and Muller-Hoissen F 2015 Simplex and polygon equations SIGMA11 042 · Zbl 1338.06001 · doi:10.3842/SIGMA.2015.042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.