Advertisement

Letters in Mathematical Physics

, Volume 104, Issue 3, pp 299–309 | Cite as

Non-Commutative Rational Yang–Baxter Maps

  • Adam Doliwa
Open Access
Article

Abstract

Starting from multidimensional consistency of non-commutative lattice-modified Gel’fand–Dikii systems, we present the corresponding solutions of the functional (set-theoretic) Yang–Baxter equation, which are non-commutative versions of the maps arising from geometric crystals. Our approach works under additional condition of centrality of certain products of non-commuting variables. Then we apply such a restriction on the level of the Gel’fand–Dikii systems what allows to obtain non-autonomous (but with central non-autonomous factors) versions of the equations. In particular, we recover known non-commutative version of Hirota’s lattice sine-Gordon equation, and we present an integrable non-commutative and non-autonomous lattice modified Boussinesq equation.

Mathematics Subject Classification (1991)

37K10 37K60 16T25 39A14 14E07 

Keywords

non-commutative integrable difference equations functional Yang–Baxter equation non-commutative rational maps non-autonomous lattice Gel’fand–Dikii systems multidimensional consistency 

References

  1. 1.
    Adler V.E., Bobenko A.I., Suris Yu.B.: Classification of integrable equations on quadgraphs. The consistency approach. Commun. Math. Phys. 233, 513–543 (2003)ADSMATHMathSciNetGoogle Scholar
  2. 2.
    Adler V.E., Bobenko A.I., Suris Yu.B.: Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings. Comm. Anal. Geom. 12, 967–1007 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)MATHGoogle Scholar
  4. 4.
    Bobenko A.I., Suris Yu.B.: Integrable non-commutative equations on quad-graphs. The consistency approach. Lett. Math. Phys. 61, 241–254 (2002)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cohn P.M.: Skew fields. Theory of general division rings. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  6. 6.
    Date E., Jimbo M., Miwa T.: Method for generating discrete soliton equations. II. J. Phys. Soc. Japan 51, 4125–4131 (1982)ADSCrossRefGoogle Scholar
  7. 7.
    Di Francesco P., Kedem R.: Discrete non-commutative integrability: Proof of a conjecture by M. Kontsevich. Int. Math. Res. Notes 2010, 4042–4063 (2010)MATHMathSciNetGoogle Scholar
  8. 8.
    Doliwa A.: Desargues maps and the Hirota–Miwa equation. Proc. R. Soc. A 466, 1177–1200 (2010)ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Doliwa, A.: Non-commutative lattice modified Gel’fand–Dikii systems. J. Phys. A: Math. Theor. 46, 205202 (2013)Google Scholar
  10. 10.
    Doliwa, A.: Desargues maps and their reductions. In: Proceedings of the 2nd International Workshop on Nonlinear and Modern Mathematical Physics (March 2013, Tampa FL), American Institute of Physics, arXiv:1307.8294 (to appear, 2013)Google Scholar
  11. 11.
    Doliwa, A., Sergeev, S.M.: The pentagon relation and incidence geometry, arXiv: 1108.0944Google Scholar
  12. 12.
    Drinfeld, V.G.: On some unsolved problems in quantum group theory. In: Kulish, P.P. (ed.) Quantum Groups (Leningrad, 1990). Lect. Notes Math. vol. 1510, pp. 1–8. Springer, Berlin (1992)Google Scholar
  13. 13.
    Etingof P.: Geometric crystals and set-theoretical solutions to the quantum Yang–Baxter equation. Comm. Algebra 31, 1961–1973 (2003)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Hirota R.: Nonlinear partial difference equations. I. A difference analog of the Korteweg–de Vries equation. J. Phys. Soc. Jpn. 43, 1423–1433 (1977)ADSGoogle Scholar
  15. 15.
    Hirota R.: Nonlinear partial difference equations. III. Discrete sine-Gordon equation. J. Phys. Soc. Jpn. 43, 2079–2086 (1977)ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hirota R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50, 3785–3791 (1981)ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    Humphreys J.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  18. 18.
    Kajiwara K., Noumi M., Yamada Y.: Discrete dynamical systems with \({W(A_{m-1}^{(1)} \times A_{n-1}^{(1)})}\) symmetry. Lett. Math. Phys. 60, 211–219 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Kajiwara K., Noumi M., Yamada Y.: q-Painlevé systems arising from q-KP hierarchy. Lett. Math. Phys. 62, 259–268 (2002)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Korepin V.E., Bogoliubov N.M., Izergin A.G.: Quantum inverse scattering method and correlation functions. Cambridge University Press, Cambridge (1993)CrossRefMATHGoogle Scholar
  21. 21.
    Kuniba, A., Nakanishi, T., Suzuki, J.: T-systems and Y-systems in integrable systems. J. Phys. A: Math. Theor. 44, 103001 (2011)Google Scholar
  22. 22.
    Kupershmidt, B.: KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems. AMS, Providence (2000)Google Scholar
  23. 23.
    Nijhoff, F.W.: Discrete Painlevé equations and symmetry reduction on the lattice. In: Bobenko, A.I., Seiler, R. (eds.) Discrete Integrable Geometry and Physics, pp. 209–234. Clarendon Press, Oxford (1999)Google Scholar
  24. 24.
    Nijhoff F.W.: Lax pair for the Adler (lattice Krichever–Novikov) system. Phys. Lett. A 297, 49–58 (2002)ADSCrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Nijhoff F.W., Capel H.W.: The direct linearization approach to hierarchies of integrable PDEs in 2 + 1 dimensions: I. Lattice equations and the differential-difference hierarchies. Inverse Problems 6, 567–590 (1990)ADSCrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Nimmo J.J.C.: On a non-Abelian Hirota-Miwa equation. J. Phys. A: Math. Gen. 39, 5053–5065 (2006)ADSCrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Papageorgiou, V.G., Tongas, A.G., Veselov, A.P.: Yang–Baxter maps and symmetries of integrable equations on quad-graphs. J. Math. Phys. 47, 083502 (2006)Google Scholar
  28. 28.
    Suris, Yu.B., Veselov, A.P.: Lax matrices for Yang–Baxter maps. J. Nonlinear Math. Phys. 10(Suppl 2), 223–230 (2003)Google Scholar
  29. 29.
    Veselov A.P.: Yang–Baxter maps and integrable dynamics. Phys. Lett. A 314, 214–221 (2003)ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of Warmia and Mazury in OlsztynOlsztynPoland

Personalised recommendations