Geodesic
Combinatorial Yang–Baxter maps arising from the tetrahedron equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 1, pp. 84-100 Cet article a éte moissonné depuis la source Math-Net.Ru

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We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum $\mathscr R$-matrices of generalized quantum groups interpolating the symmetric tensor representations of $U_q(A^{(1)}_{n-1})$ and the antisymmetric tensor representations of $U_{-q^{-1}}(A^{(1)}_{n-1})$. We show that at $q=0$, they all reduce to the Yang–Baxter maps called combinatorial $\mathscr R$-matrices and describe the latter by an explicit algorithm.
Mots-clés : tetrahedron equation
Keywords: Yang–Baxter equation, generalized quantum group, Yang–Baxter map. 
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A. Kuniba. Combinatorial Yang–Baxter maps arising from the tetrahedron equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 1, pp. 84-100. http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a6/

[1] A. B. Zamolodchikov, ZhETF, 79:2 (1980), 641–664 | MR

[2] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Dover, Mineola, NY, 2007 | MR | Zbl

[3] V. V. Bazhanov, S. M. Sergeev, J. Phys. A: Math. Theor., 39:13 (2013), 3295–3310 | DOI | MR

[4] A. Kuniba, S. Sergeev, Commun. Math. Phys., 324:3 (2013), 695–713 | DOI | MR | Zbl

[5] A. Kuniba, M. Okado, Commun. Math. Phys., 334:3 (2013), 1219–1244 | DOI | MR

[6] A. Kuniba, M. Okado, S. Sergeev, J. Phys. A: Math. Theor., 48:30 (2013), 304001, 38 pp. | DOI | MR

[7] M. M. Kapranov, V. A. Voevodsky, “$2$-categories and Zamolodchikov tetrahedron equations”, Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (Pennsylvania State University, University Park, PA, USA, July 6–26, 1991), Proceedings of Symposia in Pure Mathematics, 56, eds. W. J. Haboush, B. J. Parshall, AMS, Providence, RI, 1994, 177–259 | DOI | MR | Zbl

[8] I. Heckenberger, J. Algebra, 323:8 (2010), 2130–2182 | DOI | MR | Zbl

[9] I. Heckenberger, H. Yamane, Rev. Unión Mat. Argentina, 51:2 (2010), 107–146 | MR | Zbl

[10] V. G. Drinfel'd, “Quantum groups”, Proceedings of the International Congress of Mathematicians (ICM) (Berkeley, CA, August 3–11, 1986), ed. A. M. Gleason, AMS, Providence, RI, 1987, 798–820 | MR | Zbl

[11] M. Jimbo, Lett. Math. Phys., 10:1 (1985), 63–69 | DOI | MR | Zbl

[12] A. Nakayashiki, Y. Yamada, Selecta Math. (N. S.), 3:4 (1997), 547–599 | DOI | MR | Zbl

[13] K. Hikami, R. Inoue, J. Phys. A: Math. Gen., 33:22 (2000), 4081–4094 | DOI | MR | Zbl

[14] M. Kashiwara, Duke Math. J., 63:2 (1991), 465–516 | DOI | MR | Zbl

[15] E. Date, M. Jimbo, A. Kuniba, T. Miwa, M. Okado, Lett. Math. Phys., 17:1 (1989), 69–77 | DOI | MR | Zbl

[16] S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Internat. J. Modern Phys. A, 7, suppl. 1A (1992), 449–484 | DOI | MR | Zbl

[17] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Y. Yamada, “Remarks on fermionic formula”, Recent Developments in Quantum Affine Algebras and Related Topics (North Carolina State University, Raleigh, NC, USA, May 21–24, 1998), Contemporary Mathematics, 248, eds. N. Jing, K. C. Misra, AMS, Providence, RI, 1999, 243–291 | DOI | MR | Zbl

[18] M. Okado, “$X=M$ conjecture”, Combinatorical Aspect of Integrable Systems (RIMS, Kyoto, Japan, July 26–30, 2004), Mathematical Society of Japan Memoirs, 17, eds. A. Kuniba, M. Okado, Math. Soc. Japan, Tokyo, 2007, 43–73 | DOI | MR | Zbl

[19] R. Inoue, A. Kuniba, T. Takagi, J. Phys. A: Math. Theor., 45:7 (2012), 073001, 64 pp. | DOI | MR | Zbl

[20] A. Kuniba, Bethe Ansatz and Combinatorics, Asakura, Tokyo, 2011 (in Japanese)

[21] V. G. Drinfeld, “On some unsolved problems in quantum group theory”, Quantum Groups (Euler International Mathematical Institute, Leningrad, 1990), Lecture Notes in Mathematics, 1510, ed. P. P. Kulish, Springer, Berlin, 1992, 1–8 | DOI | MR | Zbl

[22] A. Veselov, “Yang–Baxter maps: Dynamical point of view”, Combinatorical Aspect of Integrable Systems (RIMS, Kyoto, Japan, July 26–30, 2004), Mathematical Society of Japan Memoirs, 17, eds. A. Kuniba, M. Okado, Math. Soc. Japan, Tokyo, 2007, 145–167 | DOI | MR | Zbl

[23] A. Kuniba, M. Okado, J. Phys. A: Math. Theor., 45:46 (2012), 465206, 27 pp. | DOI | MR | Zbl

[24] A. Kuniba, S. Maruyama, J. Phys. A: Math. Theor., 48:13 (2015), 135204, 19 pp. | DOI | MR | Zbl

[25] S. M. Sergeev, J. Phys. A: Math. Theor., 42:29 (2009), 295206, 19 pp. | DOI | MR | Zbl

[26] G. Benkart, S.-J. Kang, M. Kashiwara, J. Amer. Math. Soc., 13 (2000), 295–331 | DOI | MR | Zbl