From braces to Hecke algebras and quantum groups. (English) Zbl 1533.16059
Braces were introduced by
W. Rump [J. Algebra 307, No. 1, 153–170 (2007; Zbl 1115.16022)] to describe all finite involutive set-theoretic solutions of the Yang-Baxter equation. Since they are involutive, according to M. Jimbo [Commun. Math. Phys. 102, 537–547 (1986; Zbl 0604.58013)], it is possible to Baxterize them and obtain solutions to the parameter-dependent Yang-Baxter equation, which appears in quantum integrable systems. In this paper, the authors investigate connections between the theory of braces and selected topics from the theory of quantum integrable systems. In particular, they make connections with Hecke algebras and identify new quantum groups associated to solutions coming from braces. They also construct a new class of quantum discrete integrable systems and derive symmetries for the corresponding periodic transfer matrices.
Reviewer: Marzia Mazzotta (Lecce)
MSC:
| 16T20 | Ring-theoretic aspects of quantum groups |
| 16T25 | Yang-Baxter equations |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
References:
| [1] | Adler, V. E., Bobenko, A. I. and Suris, Y. B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys.233 (2003) 513. · Zbl 1075.37022 |
| [2] | Bachiller, D., Counterexample to a conjecture about braces, J. Algebra453 (2016) 160-176. · Zbl 1338.16022 |
| [3] | Bachiller, D., Cedó, F., Jespers, E. and Okniński, J., Iterated matched products of finite braces and simplicity; New solutions of the Yang-Baxter equation, Trans. Amer. Math. Soc.370 (2018) 4881-4907. · Zbl 1431.16035 |
| [4] | Baxter, R. J., Exactly Solved Models in Statistical Mechanics (Academic Press, 1982). · Zbl 0538.60093 |
| [5] | Bazhanov, V. V. and Sergeev, S. M., Yang-Baxter maps, discrete integrable equations and quantum groups, Nuclear Phys. B926 (2018) 509-543. · Zbl 1380.81164 |
| [6] | Berenstein, A. and Kazhdan, D., Geometric and unipotent crystals, in Visions in Mathematics \(:\) GAFA 2000 Special Volume (Birkhäuser, 2000), p. 188. · Zbl 1044.17006 |
| [7] | Brzezinski, T., Trusses: Between braces and ring, Trans. Amer. Math. Soc.372 (2019) 4149-4176. · Zbl 1471.16053 |
| [8] | Catino, F., Colazzo, I. and Stefanelli, P., Semi-braces and the Yang-Baxter equation, J. Algebra483 (2017) 163-187. · Zbl 1385.16035 |
| [9] | Cedó, F., Left braces: Solutions of the Yang-Baxter equation, Adv. Group Theory Appl.5 (2018) 33-90. · Zbl 1403.16033 |
| [10] | Cedó, F., Jespers, E. and Okninski, J., Braces and the Yang-Baxter equation, Comm. Math. Phys.327(1) (2014) 101-116. · Zbl 1287.81062 |
| [11] | Doikou, A., Murphy elements from the double-row transfer matrix, J. Stat. Mech., Theory Exp.2009 (2009) L03003. · Zbl 1459.82067 |
| [12] | Doikou, A. and Nepomechie, R. I., Bulk and boundary S-matrices for the \(SU(N)\) chain, Nuclear Phys. B521 (1998) 547-572. · Zbl 1047.82512 |
| [13] | Doikou, A. and Smoktunowicz, A., Set-theoretical Yang-Baxter and reflection equations & quantum group symmetries, Lett. Math. Phys.111 (2021) 105. · Zbl 1486.16039 |
| [14] | V. G. Drinfel’d, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl.32 (1985) 254; A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl.36 (1988) 212. · Zbl 0667.16004 |
| [15] | Drinfeld, V. G., On some Unsolved Problems in Quantum Group Theory, , Vol. 1510 (Springer-Verlag, Berlin, 1992), pp. 1-8. · Zbl 0765.17014 |
| [16] | Etingof, P., Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Comm. Algebra31 (2001) 1961-1973. · Zbl 1020.17008 |
| [17] | Etingof, P., Schedler, T. and Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J.100 (1999) 169-209. · Zbl 0969.81030 |
| [18] | Faddeev, L. D., Reshetikhin, N. Y. and Takhtajan, L. A., Quantization of Lie groups and Lie algebras, Leningrad Math. J.1 (1990) 193. · Zbl 0715.17015 |
| [19] | Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987). · Zbl 1111.37001 |
| [20] | Gateva-Ivanova, T., Skew polynomial rings with binomial relations, J. Algebra185 (1996) 710-753. · Zbl 0863.16016 |
| [21] | Gateva-Ivanova, T., Binomial skew polynomial rings, Artin-Schelter regularity, and binomial solutions of the Yang-Baxter equation, Serdica Math. J.30(2-3) (2004) 431-470. · Zbl 1066.16026 |
| [22] | Gateva-Ivanova, T., Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity, Adv. Math.230 (2012) 2152-2175. · Zbl 1267.81209 |
| [23] | Gateva-Ivanova, T., Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math.388(7) (2018) 649-701. · Zbl 1437.16028 |
| [24] | Gateva-Ivanova, T., A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation, Publ. Mat.65(2) (2021) 747-808. · Zbl 1491.16035 |
| [25] | Gateva-Ivanova, T. and Van den Bergh, M., Semigroups of I-type, J. Algebra206 (1997) 97-112. · Zbl 0944.20049 |
| [26] | Guarnieri, L. and Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comp.86(307) (2017) 2519-2534. · Zbl 1371.16037 |
| [27] | Hatayama, G., Kuniba, A. and Takagi, T., Soliton cellular automata associated with crystal bases, Nuclear Phys. B577 (2000) 619. · Zbl 1024.82017 |
| [28] | Hietarinta, J., Permutation-type solutions to the Yang-Baxter and other n-simplex equations, J. Phys. A30 (1997) 4757-4771. · Zbl 0939.81012 |
| [29] | Jedlicka, P., Pilitowska, A. and Zamojska-Dzienio, A., The retraction relation for biracks, J. Pure Appl. Algebra223 (2019) 3594-3610. · Zbl 1411.16032 |
| [30] | Jespers, E., Kubat, E. and Van Antwerpen, A., The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation, Trans. Amer. Math. Soc.372 (2019) 7191-7223. · Zbl 1432.16032 |
| [31] | Jespers, E., Kubat, E., Van Antwerpen, A. and Vendramin, L., Factorizations of skew braces, Math. Ann.375(3-4) (2019) 1649-1663. · Zbl 1446.16040 |
| [32] | Jespers, E. and Okniński, J., Binomial semigroups, J. Algebra202 (1998) 250-275. · Zbl 0910.20038 |
| [33] | Jespers, E. and Okniński, J., Monoids and groups of I-type, Algebr. Represent. Theory8 (2005) 709-729. · Zbl 1091.20024 |
| [34] | Jespers, E. and Okniński, J., Noetherian Semigroup Algebras, , Vol. 7 (Springer, 2007). · Zbl 1135.16001 |
| [35] | Jespers, E., Okniński, J. and Van Campenhout, M., Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids, J. Algebra440 (2015) 72-99. · Zbl 1346.16024 |
| [36] | Jespers, E. and Van Campenhout, M., Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids II, J. Algebra492 (2017) 524-546. · Zbl 1379.16017 |
| [37] | Jimbo, M., A q-difference analogue of \(U(g)\) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985) 63. · Zbl 0587.17004 |
| [38] | Jimbo, M., Quantum R-matrix for the generalized Toda system, Comm. Math. Phys.102 (1986) 537-547. · Zbl 0604.58013 |
| [39] | Kauffman, L. H., Virtual knot theory, European J. Combin.20 (1999) 663-691. · Zbl 0938.57006 |
| [40] | Konovalov, A., Smoktunowicz, A. and Vendramin, L., On skew braces and their ideals, Experiment. Math.30 (2021) 95-104. · Zbl 1476.16036 |
| [41] | Lau, I., An associative left brace is a ring, J. Algebra Appl.19(9) (2020) 2050179. · Zbl 1467.16037 |
| [42] | Lebed, V. and Vendramin, L., On structure groups of set-theoretical solutions to the Yang-Baxter equation, Proc. Edinb. Math. Soc.62(3) (2019) 683-717. · Zbl 1423.16034 |
| [43] | Molev, A., Nazarov, M. and Olshanski, G., Yangians and classical Lie algebras, Russian Math. Surveys51 (1996) 205. · Zbl 0876.17014 |
| [44] | Papageorgiou, V. G., Suris, Y. B., Tongas, A. G. and Veselov, A. P., On quadrirational Yang-Baxter maps, SIGMA6 (2010) 33. · Zbl 1190.14012 |
| [45] | Rump, W., A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math.193(1) (2005) 40-55. · Zbl 1074.81036 |
| [46] | Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra307(1) (2007) 153-170. · Zbl 1115.16022 |
| [47] | Smoktunowicz, A. and Smoktunowicz, A., Set-theoretic solutions of the Yang-Baxter equation and new classes of \(R\)-matrices, Linear Algebra Appl.546 (2018) 86-114. · Zbl 1384.16025 |
| [48] | Smoktunowicz, A. and Vendramin, L., On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra2(1) (2018) 47-86. · Zbl 1416.16037 |
| [49] | Y. P. Sysak, The adjoint group of radical rings and related questions, in Ischia Group Theory 2010 (World Scientific, Singapore, 2011), pp. 344-365; Proc. Conf., Ischia, Naples, Italy, 14-17 April 2010. · Zbl 1298.16010 |
| [50] | Takahashi, D. and Satsuma, J., A soliton cellular automaton, J. Phys. Soc. Japan59 (1990) 3514. · Zbl 0885.68112 |
| [51] | Vendramin, L., Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra220(5) (2016) 1681-2076. · Zbl 1337.16028 |
| [52] | Veselov, A. P., Yang-Baxter maps and integrable dynamics, Phys. Lett. A314 (2003) 214. · Zbl 1051.81014 |
| [53] | Yang, C. N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Rev. Lett.19 (1967) 1312. · Zbl 0152.46301 |
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