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Combinatorial Yang–Baxter maps arising from the tetrahedron equation

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Abstract

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of generalized quantum groups interpolating the symmetric tensor representations of U q (A n−1 (1)) and the antisymmetric tensor representations of \({U_{ - {q^{ - 1}}}}\left( {A_{n - 1}^{\left( 1 \right)}} \right)\) . We show that at q = 0, they all reduce to the Yang–Baxter maps called combinatorial R-matrices and describe the latter by an explicit algorithm.

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Authors and Affiliations

  1. University of Tokyo, Tokyo, Japan

    A. Kuniba

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  1. A. Kuniba
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Correspondence to A. Kuniba.

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This research is supported by Grants-in-Aid for Scientific Research No. 15K13429.

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 1, pp. 84–100, October, 2016.

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Kuniba, A. Combinatorial Yang–Baxter maps arising from the tetrahedron equation. Theor Math Phys 189, 1472–1485 (2016). https://doi.org/10.1134/S004057791610007X

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