Hom-Yang-Baxter equations and Hom-Yang-Baxter systems. (English) Zbl 1525.16033
The authors of the paper present examples of solutions to a generalization of the Yang-Baxter equation called the Hom-Yang-Baxter equation (defined in [D. Yau, J. Phys. A, Math. Theor. 42, No. 16, Article ID 165202, 12 p. (2009; Zbl 1179.17001)])
In particular, they construct such solutions (and their inverses) from Hom-algebras, Hom-co algebras, and Hom-Lie algebras.
Finally, they generalize the Yang-Baxter system to the Hom-Yang-Baxter system and they give some examples.
In particular, they construct such solutions (and their inverses) from Hom-algebras, Hom-co algebras, and Hom-Lie algebras.
Finally, they generalize the Yang-Baxter system to the Hom-Yang-Baxter system and they give some examples.
Reviewer: Agata Pilitowska (Warszawa)
MSC:
| 16T25 | Yang-Baxter equations |
| 17A30 | Nonassociative algebras satisfying other identities |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
Citations:
Zbl 1179.17001References:
| [1] | Baxter, R., Partition function of the eight-vertex lattice model, Ann. Phys, 70, 1, 193-228 (1972) · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1 |
| [2] | Berceanu, B. R.; Nichita, F. F.; Popescu, C., Algebra structures arising from Yang-Baxter systems, Commun. Algebra, 41, 12, 4442-4452 (2013) · Zbl 1300.16035 · doi:10.1080/00927872.2012.703736 |
| [3] | Brzeziński, T.; Nichita, F. F., Yang-Baxter systems and entwining structures, Commun. Algebra, 33, 4, 1083-1093 (2005) · Zbl 1085.16028 · doi:10.1081/AGB-200053815 |
| [4] | Chen, J.; Xue, K.; Ge, M., Berry phase and quantum criticality in Yang-Baxter systems, Ann. Phys, 323, 10, 2614-2623 (2008) · Zbl 1192.81176 · doi:10.1016/j.aop.2008.06.003 |
| [5] | Chen, Y.; Wang, Z.; Zhang, L., Quasitriangular Hom-Hopf algebras, Colloq. Math, 137, 1, 67-88 (2014) · Zbl 1336.16031 · doi:10.4064/cm137-1-5 |
| [6] | Chen, Y.; Zhang, L., The category of Yetter-Drinfel’d Hom-modules and the quantum Hom-Yang-Baxter equation, J. Math. Phys, 55, 3, 031702 (2014) · Zbl 1292.16022 · doi:10.1063/1.4868964 |
| [7] | Chen, Y.; Zhou, X., Separable and Frobenius monoidal Hom-algebras, Colloq. Math, 137, 2, 229-251 (2014) · Zbl 1336.16032 · doi:10.4064/cm137-2-8 |
| [8] | Fang, X.; Liu, W., Solutions of the BiHom-Yang-Baxter equation, Sb. Math, 209, 6, 901-918 (2018) · Zbl 1442.16035 · doi:10.1070/SM8863 |
| [9] | Gohr, A., On hom-algebras with surjective twisting, J. Algebra, 324, 7, 1483-1491 (2010) · Zbl 1236.17003 · doi:10.1016/j.jalgebra.2010.05.003 |
| [10] | Harrathi, F.; Mabrouk, S.; Ncib, O.; Silvestrov, S., Malcev Yang-Baxter equation, weighted O-operators on Malcev algebras and post-Malcev algebras, arXiv: 2206.03629v1 (2022) |
| [11] | Hartwig, J.; Larsson, D.; Silvestrov, S., Deformations of Lie algebras using σ-derivations, J. Algebra, 295, 2, 314-361 (2006) · Zbl 1138.17012 · doi:10.1016/j.jalgebra.2005.07.036 |
| [12] | Hlavaty, L.; Snobl, L., Solution of the Yang-Baxter system for quantum doubles, Int. J. Mod. Phys. A, 14, 19, 3029-3058 (1999) · Zbl 0957.17015 · doi:10.1142/S0217751X99001470 |
| [13] | Hu, N., q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebr. Colloq, 6, 1, 51-70 (1999) · Zbl 0943.17007 · doi:10.1007/s10114-999-0090-4 |
| [14] | Li, H.; Ma, T., A construction of the Hom-Yetter-Drinfeld category, Colloq. Math, 137, 1, 43-65 (2014) · Zbl 1306.16031 · doi:10.4064/cm137-1-4 |
| [15] | Liu, L.; Shen, B., Radford’s biproducts and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras, J. Math. Phys, 55, 3, 031701 (2014) · Zbl 1306.16032 · doi:10.1063/1.4866760 |
| [16] | Lu, D.; Li, Y.; Guo, S., Crossed products of Hom-Hopf algebras, Filomat, 34, 4, 1295-1313 (2020) · Zbl 1499.16083 · doi:10.2298/FIL2004295L |
| [17] | Ma, T.; Li, H.; Yang, T., Cobraided smash product Hom-Hopf algebras, Colloq. Math, 134, 1, 75-92 (2014) · Zbl 1304.16034 · doi:10.4064/cm134-1-3 |
| [18] | Ma, T.; Wang, Y.; Liu, L., Generalized Radford biproduct Hom-Hopf algebras and related braided tensor categories, J. Math, 37, 6, 1161-1172 (2017) · Zbl 1399.16082 · doi:10.13548/j.sxzz.20160329.001 |
| [19] | Ma, T.; Yang, H.; Liu, L.; Chen, Q., On unified Hom-Yetter-Drinfeld categories, J. Geom. Phys, 144, 7, 81-107 (2019) · Zbl 1462.17023 · doi:10.1016/j.geomphys.2019.05.015 |
| [20] | Makhlouf, A.; Silvestrov, S., Hom-algebra structures, J. Gen. Lie Theory Appl., 2, 2, 51-64 (2008) · Zbl 1184.17002 · doi:10.4303/jglta/S070206 |
| [21] | Makhlouf, A.; Silvestrov, S.; Silvestrov, S.; Paal, E.; Abramov, V.; Stolin, A., Generalized Lie Theory in Mathematics, Physics and Beyond, Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, 189-206 (2009), Berlin: Springer-Verlag, Berlin · Zbl 1173.16019 · doi:10.1007/978-3-540-85332-9-17 |
| [22] | Makhlouf, A.; Silvestrov, S., Hom-algebras and Hom-coalgebras, J. Algebra Appl., 9, 4, 553-589 (2010) · Zbl 1259.16041 · doi:10.1142/S0219498810004117 |
| [23] | Makhlouf, A.; Panaite, F., Yetter-Drinfeld modules for Hom-bialgebras, J. Math. Phys, 55, 1, 013501 (2014) · Zbl 1292.16025 · doi:10.1063/1.4858875 |
| [24] | Mishra, S. K.; Naolekar, A., \(####\)-operators on Hom-Lie algebras, J. Math. Phys, 61, 121701 (2020) · Zbl 1477.17070 · doi:10.1063/5.0026719 |
| [25] | Nichita, F., Self-inverse Yang-Baxter operators from (co)algebra structures, J. Algebra, 218, 2, 738-759 (1999) · Zbl 0944.16033 · doi:10.1006/jabr.1999.7915 |
| [26] | Nichita, F.; Parashar, D., Spectral-parameter dependent Yang-Baxter operators and Yang-Baxter systems from algebra structures, Commun. Algebra, 34, 8, 2713-2726 (2006) · Zbl 1099.81039 · doi:10.1080/00927870600651661 |
| [27] | Nichita, F.; Popovici, B. P., Some results on the Yang-Baxter equations and applications, Romanian J. Phys, 53, 9-10, 1177-1182 (2008) · Zbl 1187.81160 |
| [28] | Nichita, F.; Parashar, D., English summary) Lie algebras, Vertex Operator Algebras and their Applications, Contemporary Mathamatics, 442, New constructions of Yang-Baxter systems, 193-200 (2007), Providence, RI: American Mathematical Society, Providence, RI |
| [29] | Sweedler, M. E., Hopf Algebras (1969), New York: Benjamin, New York · Zbl 0194.32901 |
| [30] | Wang, S.; Guo, S., Symmetries and the u-condition in Hom-Yetter-Drinfeld categories, J. Math. Phys, 55, 8, 081708 (2014) · Zbl 1306.16034 · doi:10.1063/1.4892081 |
| [31] | Wang, S.; Guo, S., Symmetric pairs and pseudosymmetries in Hom-Yetter-Drinfeld categories, J. Algebra Appl., 16, 7, 1750125 (2017) · Zbl 1385.16033 · doi:10.1142/S0219498817501250 |
| [32] | Wang, S.; Guo, S., BiHom-Lie superalgebra structures and BiHom-Yang-Baxter equations, Adv. Appl. Clifford Algebras, 30, 3, 18 (2020) · Zbl 1476.17019 · doi:10.1007/s00006-020-01060-0 |
| [33] | Wang, S.; Zhang, X.; Guo, S., The Hom-Long dimodule category and nonlinear equations, Electron. Res. Arch, 30, 1, 362-381 (2022) · Zbl 1494.81010 · doi:10.3934/era.2022019 |
| [34] | Wang, Z.; Chen, Y.; Zhang, L., The antipode and Drinfel’d double of Hom-Hopf algebras, Sci. Sin. Math, 42, 11, 1079-1093 (2012) · Zbl 1488.16090 · doi:10.1360/012011-138 |
| [35] | Yang, C. N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19, 1312-1315 (1967) · Zbl 0152.46301 · doi:10.1103/PhysRevLett.19.1312 |
| [36] | Yau, D., The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras, J. Phys. A, 42, 16, 165202 (2009) · Zbl 1179.17001 · doi:10.1088/1751-8113/42/16/165202 |
| [37] | Yau, D., The Hom-Yang-Baxter equation and Hom-Lie algebras, J. Math. Phys, 52, 5, 053502 (2011) · Zbl 1317.16032 · doi:10.1063/1.3571970 |
| [38] | Yau, D., Hom-quantum groups I: Quasi-triangular Hom-bialgebras, J. Phys. A, 45, 6, 065203 (2012) · Zbl 1241.81110 · doi:10.1088/1751-8113/45/6/065203 |
| [39] | Yau, D., The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras, Int. Electron. J. Algebra, 17, 17, 11-45 (2015) · Zbl 1323.16027 · doi:10.24330/ieja.266210 |
| [40] | You, M.; Wang, S., Constructing new braided T-categories over monoidal Hom-Hopf algebras, J. Math. Phys, 55, 11, 111701 (2014) · Zbl 1307.81061 · doi:10.1063/1.4900824 |
| [41] | Zhang, X.; Guo, S.; Wang, S., Drinfeld codoubles of Hom-Hopf algebras, Adv. Appl. Clifford Algebras, 29, 2, 26 (2019) · Zbl 1454.17010 · doi:10.1007/s00006-019-0949-0 |
| [42] | Zhang, X.; Wang, W.; Zhao, X., Smash coproducts of monoidal comonads and Hom-entwining structures, Rocky Mountain J. Math, 49, 6, 2063-2105 (2019) · Zbl 1511.16031 · doi:10.1216/RMJ-2019-49-6-2063 |
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