×

Set-theoretic solutions of the Yang-Baxter equation and new classes of \(\mathrm{R}\)-matrices. (English) Zbl 1384.16025

Summary: We describe several methods of constructing \(R\)-matrices that are dependent upon many parameters, for example unitary \(R\)-matrices and \(\mathrm{R}\)-matrices whose entries are functions. As an application, we construct examples of \(R\)-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang-Baxter equation (QYBE) and construct \(R\)-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such solutions are abundant. We show that \(R\)-matrices related to involutive, non-degenerate solutions of the QYBE have special form. We also investigate some linear algebra questions related to \(R\)-matrices.

MSC:

16T25 Yang-Baxter equations
15A69 Multilinear algebra, tensor calculus
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Dancer, K. A.; Finch, P. E.; Isaac, P. S., Universal baxterisation for \(Z\)-graded Hopf algebras, J. Phys. A, 40, 50, 1069-1075 (2007) · Zbl 1134.81383
[2] Kauffman, L. H.; Lomonaco, S. J., Braiding operators are universal quantum gates, New J. Phys., 6, 134 (2004)
[3] Nayak, C.; Simon, S. H.; Stern, A.; Freedman, M.; Sarma, S. D., Non-Abelian anyons and topological quantum computation, Rev. Modern Phys., 80 (2008) · Zbl 1205.81062
[4] Galindo, C.; Rowell, E. C., Braid representations from unitary braided vector spaces, J. Math. Phys., 55, Article 061702 pp. (2014) · Zbl 1335.20040
[5] Franko, J. M., Braid group representations arising from the Yang-Baxter equation, J. Knot Theory Ramifications, 19, 525 (2010) · Zbl 1211.20031
[6] Brzeziński, T.; Nichita, F. F., Yang-Baxter systems and entwined structures, Comm. Algebra, 33, 1083-1093 (2005) · Zbl 1085.16028
[7] Kharchenko, V., Quantum Lie Theory, a Multilinear Approach, Lecture Notes in Mathematics, vol. 2150 (2015), Springer International Publishing · Zbl 1337.17001
[8] Larsen, M.; Rowell, E. C., Unitary braid representations with finite image, Algebr. Geom. Topol., 8, 2063-2079 (2008) · Zbl 1187.20047
[9] Iordanescu, R.; Nichita, F. F.; Nichita, I. M., The Yang-Baxter equation, (quantum) computers and unifying theories, Axioms, 3, 360-368 (2014) · Zbl 1387.81158
[10] Etingof, P.; Schedler, T.; Soloviev, T., A set theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 169-209 (1999) · Zbl 0969.81030
[11] Etingof, P.; Gelaki, S., A method of construction of finite-dimensional triangular semisimple Hopf algebras, Math. Res. Lett., 5, 551-561 (1998) · Zbl 0935.16029
[12] Chen, R. S., Generalized Yang-Baxter equations and braiding quantum gates, J. Knot Theory Ramifications, 21, 9 (2012) · Zbl 1247.81084
[13] Rowell, E. C.; Wang, Z., Localization of unitary braid group representations, Comm. Math. Phys., 3, 595-615 (2012) · Zbl 1245.81023
[14] E.C. Rowell, private communication, February 2017.; E.C. Rowell, private communication, February 2017.
[15] Cui, S. X.; Hong, S. M.; Wang, Z., Universal quantum computation with weakly integral anyons, Quantum Inf. Process., 14, 8, 2687-2727 (2015) · Zbl 1327.81127
[16] Andruskiewitsch, N.; Grańa, M., From racks to pointed Hopf algebras, Adv. Math., 178, 2, 177-243 (2003) · Zbl 1032.16028
[17] Galindo, C.; Hong, S. M.; Rowell, E. C., Generalized and quasi-localization of braid group representations, Int. Math. Res. Not. IMRN, 3, 693-731 (2013) · Zbl 1318.20037
[18] Rowell, E. C., Parameter dependent Gaussian (z, N)-generalized Yang-Baxter operators, Quantum Inf. Comput., 16, 1,2, Article 0105 pp. (2016)
[19] Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 153-170 (2007) · Zbl 1115.16022
[20] Carter, J. S.; Jelsovsky, D.; Kamada, S.; Saito, M., Quandle homology groups, their Betti numbers, and virtual knots, J. Pure Appl. Algebra, 157, 2-3, 135-155 (2001) · Zbl 0977.55013
[21] Lebed, V.; Vendramin, L., Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. Math., 304, 1219-1261 (2017) · Zbl 1356.16027
[22] Etingof, P.; Guralnick, R.; Soloviev, A., Indecomposable set-theoretical solutions to the quantum Yang-Baxter equation on a set with a prime number of elements, J. Algebra, 249, 709-719 (2001) · Zbl 1018.17007
[23] Klimyk, A.; Schmudgen, K., Quantum Groups and Their Representations (1997), Springer · Zbl 0891.17010
[24] Carter, J. S.; Elhamdadi, M.; Saito, M., Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles, Fund. Math., 184, 31-54 (2004) · Zbl 1067.57006
[25] Vendramin, L., Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra, 220, 5, 1681-2076 (2016) · Zbl 1337.16028
[26] Bachiller, D.; Cedó, F.; Jespers, E.; Okniński, J., A family of irretractable square-free solutions of the Yang-Baxter equation, Forum Math., 29, 6, 1291-1306 (2017) · Zbl 1394.16041
[27] Catino, F.; Rizzo, R., Regular subgroups of the affine group and radical circle algebras, Bull. Aust. Math. Soc., 79, 103-107 (2009) · Zbl 1184.20001
[28] Calderini, M.; Sala, M., Elementary abelian regular subgroups as hidden sum for cryptographic trapdoors (2017)
[29] Catino, F.; Colazzo, I.; Stefanelli, P., Regular subgroups of the affine group, Bull. Aust. Math. Soc., 91, 76-85 (2015) · Zbl 1314.20001
[30] Franko, J. M., Braid Group Representations Via the Yang-Baxter Equation (2007), Indiana University, ProQuest Dissertations Publishing
[31] Kassel, C., Quantum Groups, Graduate Text in Mathematics, vol. 155 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0808.17003
[32] Gateva-Ivanova, T., Set-theoretic solutions of the Yang-Baxter equation, braces, and symmetric groups (2015) · Zbl 1437.16028
[33] Brown, K. A.; Goodearl, K. R., Lectures on Algebraic Quantum Groups, Advanced Courses in Mathematics (2002), CRB Barcelona: CRB Barcelona Birkháuster · Zbl 1027.17010
[34] Rump, W., A decomposition theorem for square-free unitary solutions of the Yang-Baxter equation, Adv. Math., 193, 40-55 (2005) · Zbl 1074.81036
[35] Cedó, F.; Jespers, E.; Okniński, J., Braces and the Yang-Baxter equation, Comm. Math. Phys., 327, 1, 101-116 (2014) · Zbl 1287.81062
[36] Cedó, F.; Gateva-Ivanova, T.; Smoktunowicz, A., On the Yang-Baxter equation and left nilpotent left braces, J. Pure Appl. Algebra, 221, 751-756 (2016) · Zbl 1397.16033
[37] Gateva-Ivanova, T., A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation, J. Math. Phys., 45, 3828-3858 (2004) · Zbl 1065.16037
[38] Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation (2016) · Zbl 1371.16037
[39] Smoktunowicz, A.; Vendramin, L., On skew braces (2017) · Zbl 1483.16037
[40] Bachiller, D.; Cedó, F.; Jespers, E., Solutions of the Yang-Baxter equation associated with a left brace, J. Algebra, 463, 80-102 (2016) · Zbl 1348.16027
[41] Bachiller, D., Extensions, matched products, and simple braces (2016) · Zbl 1437.20031
[42] Bachiller, D.; Cedó, F.; Jespers, E.; Okniński, J., Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation, Trans. Amer. Math. Soc. (2018), published electronically: February 1, 2018 · Zbl 1431.16035
[43] Smoktunowicz, A., On Engel groups, nilpotent groups, braces and the Yang-Baxter equation, Trans. Amer. Math. Soc. (2018), in press · Zbl 1440.16040
[44] J. Okniński, private communication, January 2017.; J. Okniński, private communication, January 2017.
[45] Jespers, E.; Okniński, J., Monoids and group of I-type, Algebr. Represent. Theory, 8, 709-729 (2005) · Zbl 1091.20024
[46] Jespers, E.; Okniński, J., Noetherian Semigroup Rings (2007), Springer: Springer Dordrecht · Zbl 1178.16025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.