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Lax matrices for Yang-Baxter maps. (English) Zbl 1362.39016

Summary: It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff.

MSC:

39A12 Discrete version of topics in analysis
16T25 Yang-Baxter equations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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References:

[1] Drinfeld V.G.On some unsolved problems in quantum group theory.In ”Quantum groups” (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, 1992, p. 1–8
[2] Goncharenko V.M. Veselov A.P.Yang-Baxter maps and matrix solitons.math-ph/0303032. To appear in Proceedings of NATO ARW conference (Cadiz, June 2002)
[3] Etingof P.Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter rela-tion.math.QA/0112278 · Zbl 1020.17008
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