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.
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Doliwa, A. Non-Commutative Rational Yang–Baxter Maps. Lett Math Phys 104, 299–309 (2014). https://doi.org/10.1007/s11005-013-0669-7
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DOI: https://doi.org/10.1007/s11005-013-0669-7