Schur covers of skew braces. (English) Zbl 07804819
Summary: We develop the theory of Schur covers of finite skew braces. We prove the existence of at least one Schur cover. We also compute several examples. We prove that different Schur covers are isoclinic. Finally, we prove that Schur covers have the lifting property concerning projective representations of skew braces.
MSC:
| 16T25 | Yang-Baxter equations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
Yang-Baxter; Schur multiplier; Schur cover; cohomology; skew brace; representations; projective representationReferences:
| [1] | Bachiller, D., Classification of braces of order \(p^3\), J. Pure Appl. Algebra, 219, 8, 3568-3603 (2015) · Zbl 1312.81099 |
| [2] | Bachiller, D., Extensions, matched products, and simple braces, J. Pure Appl. Algebra, 222, 7, 1670-1691 (2018) · Zbl 1437.20031 |
| [3] | Ballester-Bolinches, A.; Esteban-Romero, R., Triply factorised groups and the structure of skew left braces, Commun. Math. Stat., 10, 2, 353-370 (2022) · Zbl 1490.81094 |
| [4] | Beyl, F. R.; Tappe, J., Group Extensions, Representations, and the Schur Multiplicator, Lecture Notes in Mathematics, vol. 958 (1982), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0544.20001 |
| [5] | Bonatto, M.; Jedlička, P., Central nilpotency of skew braces, J. Algebra Appl., 22, 12, Article 2350255 pp. (2023) · Zbl 1533.16061 |
| [6] | Catino, F.; Colazzo, I.; Stefanelli, P., Skew left braces with non-trivial annihilator, J. Algebra Appl., 18, 2, Article 1950033 pp. (2019), 23 · Zbl 1447.16035 |
| [7] | Cedó, F.; Jespers, E.; del Río, Á., Involutive Yang-Baxter groups, Trans. Am. Math. Soc., 362, 5, 2541-2558 (2010) · Zbl 1188.81115 |
| [8] | Cedó, F.; Jespers, E.; Okniński, J., Braces and the Yang-Baxter equation, Commun. Math. Phys., 327, 1, 101-116 (2014) · Zbl 1287.81062 |
| [9] | Childs, L. N.; Greither, C.; Keating, K. P.; Koch, A.; Kohl, T.; Truman, P. J.; Underwood, R. G., Hopf Algebras and Galois Module Theory, Mathematical Surveys and Monographs, vol. 260 (2021), American Mathematical Society: American Mathematical Society Providence, RI, © 2021 · Zbl 1489.16001 |
| [10] | Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 2, 169-209 (1999) · Zbl 0969.81030 |
| [11] | Gateva-Ivanova, T., Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math., 338, 649-701 (2018) · Zbl 1437.16028 |
| [12] | Gateva-Ivanova, T.; Majid, S., Matched pairs approach to set theoretic solutions of the Yang-Baxter equation, J. Algebra, 319, 4, 1462-1529 (2008) · Zbl 1140.16016 |
| [13] | Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of I-type, J. Algebra, 206, 1, 97-112 (1998) · Zbl 0944.20049 |
| [14] | Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comput., 86, 307, 2519-2534 (2017) · Zbl 1371.16037 |
| [15] | Hall, P., The classification of prime-power groups, J. Reine Angew. Math., 182, 130-141 (1940) · JFM 66.0081.01 |
| [16] | Hoffman, P. N.; Humphreys, J. F., Projective representations of the symmetric groups, (Q-Functions and Shifted Tableaux, Oxford Mathematical Monographs (1992), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York), Oxford Science Publications · Zbl 0777.20005 |
| [17] | Jespers, E.; Kubat, L.; Van Antwerpen, A.; Vendramin, L., Factorizations of skew braces, Math. Ann., 375, 3-4, 1649-1663 (2019) · Zbl 1446.16040 |
| [18] | Jespers, E.; Van Antwerpen, A.; Vendramin, L., Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation, Commun. Contemp. Math., 25, 09, Article 2250064 pp. (2023) · Zbl 1533.16063 |
| [19] | Karpilovsky, G., The Schur Multiplier, London Mathematical Society Monographs. New Series, vol. 2 (1987), The Clarendon Press Oxford University Press: The Clarendon Press Oxford University Press New York · Zbl 0619.20001 |
| [20] | Lebed, V.; Vendramin, L., Cohomology and extensions of braces, Pac. J. Math., 284, 1, 191-212 (2016) · Zbl 1357.20009 |
| [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] | Letourmy, T.; Vendramin, L., Isoclinism of skew braces, Bull. Lond. Math. Soc., 55, 6, 2891-2906 (2023) · Zbl 1535.16043 |
| [23] | Lu, J.-H.; Yan, M.; Zhu, Y.-C., On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1, 1-18 (2000) · Zbl 0960.16043 |
| [24] | Rathee, N.; Yadav, M. K., Cohomology, extensions and automorphisms of skew braces, J. Pure Appl. Algebra, 228, 2, Article 107462 pp. (2024), 30 · Zbl 1529.20048 |
| [25] | Robinson, D. J.S., A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80 (1996), Springer-Verlag: Springer-Verlag New York |
| [26] | Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 1, 153-170 (2007) · Zbl 1115.16022 |
| [27] | Rump, W., Classification of cyclic braces, J. Pure Appl. Algebra, 209, 3, 671-685 (2007) · Zbl 1170.16031 |
| [28] | Rump, W., The brace of a classical group, Note Mat., 34, 1, 115-144 (2014) · Zbl 1344.14029 |
| [29] | Schur, J., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 139, 155-250 (1911) · JFM 42.0154.02 |
| [30] | Smoktunowicz, A., On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation, Trans. Am. Math. Soc., 370, 9, 6535-6564 (2018) · Zbl 1440.16040 |
| [31] | Smoktunowicz, A.; Vendramin, L., On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra, 2, 1, 47-86 (2018) · Zbl 1416.16037 |
| [32] | Soloviev, A., Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation, Math. Res. Lett., 7, 5-6, 577-596 (2000) · Zbl 1046.81054 |
| [33] | Zhu, H., The construction of braided tensor categories from Hopf braces, Linear Multilinear Algebra, 70, 16, 3171-3188 (2022) · Zbl 1510.16030 |
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.
