Letters in Mathematical Physics

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

Non-Commutative Rational Yang–Baxter Maps

  • Adam Doliwa
Open Access


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 


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


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© 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

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