×

Skew braces of squarefree order. (English) Zbl 1476.16034

The authors use the theory of Hopf-Galois extensions and, in particular, the number of non-isomorphic Hopf-Galois extensions of square-free degree to enumerate skew braces of square-free order. As an application, a conjecture on the number of isomorphism classes of skew braces of order \(2pq\) for distinct odd primes \(p\) and \(q\) is confirmed.

MSC:

16T25 Yang-Baxter equations
16Y99 Generalizations
12F10 Separable extensions, Galois theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acri, E. and Bonatto, M., Skew braces of size \(pq\), Commun. Algebra48 (2020) 1872-1881. · Zbl 1437.16027
[2] Alabdali, A. A. and Byott, N. P., Counting Hopf-Galois structures on cyclic field extensions of squarefree degree, J. Algebra493 (2018) 1-19. · Zbl 1418.12001
[3] Alabdali, A. A. and Byott, N. P., Hopf-Galois structures of squarefree degree, J. Algebra559 (2020) 58-86. · Zbl 1465.12005
[4] Bachiller, D., Classification of braces of order \(p^3\), J. Pure Appl. Algebra219(8) (2015) 3568-3603. · Zbl 1312.81099
[5] Bachiller, D., Counterexample to a conjecture about braces, J. Algebra453 (2016) 160-176. · Zbl 1338.16022
[6] V. G. Bardakov, M. V. Neshchadim and M. K. Yadav, Computing skew left braces of small orders, to appear, Internat. J. Alg. Comput.30 (2020) 839-851. · Zbl 1458.16040
[7] Caranti, A., Volta, F. Dalla and Sala, M., Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen69(3) (2006) 297-308. · Zbl 1123.20002
[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., Gateva-Ivanova, T. and Smoktunowicz, A., Braces and symmetric groups with special conditions, J. Pure Appl. Algebra222(12) (2018) 3877-3890. · Zbl 1427.16027
[10] Cedó, F., Jespers, E. and Okniński, J., Nilpotent groups of class three and braces, Publ. Mat.60(1) (2016) 55-79. · Zbl 1345.16039
[11] Chase, S. U. and Sweedler, M. E., Hopf Algebras and Galois Theory, , Vol. 97 (Springer-Verlag, Berlin-New York, 1969). · Zbl 0197.01403
[12] Childs, L. N., Skew braces and the Galois correspondence for Hopf-Galois structures, J. Algebra511 (2018) 270-291. · Zbl 1396.12003
[13] Childs, L. N., Bi-skew braces and Hopf Galois structures, New York J. Math.25 (2019) 574-588. · Zbl 1441.12001
[14] Drinfeld, V. G., On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad, 1990), , Vol. 1510, (Springer, Berlin, 1992), pp. 1-8.
[15] Etingof, P., Schedler, T. and Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J.100(2) (1999) 169-209. · Zbl 0969.81030
[16] Featherstonhaugh, S. C., Caranti, A. and Childs, L. N., Abelian Hopf-Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc.364(7) (2012) 3675-3684. · Zbl 1287.12002
[17] Greither, C. and Pareigis, B., Hopf-Galois theory for separable field extensions, J. Algebra106(1) (1987) 239-258. · Zbl 0615.12026
[18] Guarnieri, L. and Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comp.86(307) (2017) 2519-2534. · Zbl 1371.16037
[19] Lebed, V. and Vendramin, L., Cohomology and extensions of braces, Pacific J. Math.284(1) (2016) 191-212. · Zbl 1357.20009
[20] Murty, M. R. and Murty, V. K., On groups of squarefree order, Math. Ann.267(3) (1984) 299-309. · Zbl 0531.10048
[21] 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
[22] Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra307(1) (2007) 153-170. · Zbl 1115.16022
[23] Rump, W., Classification of cyclic braces, J. Pure Appl. Algebra209(3) (2007) 671-685. · Zbl 1170.16031
[24] 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
[25] Vendramin, L., Problems on skew left braces, Adv. Group Theory Appl.7 (2019) 15-37. · Zbl 1468.16050
[26] K. N. Zenouz, On Hopf-Galois structures and skew braces of order \(p^3\), Ph.D. thesis, University of Exeter (2018).
[27] Zenouz, K. N., Skew braces and Hopf-Galois structures of Heisenberg type, J. Algebra524 (2019) 187-225. · Zbl 1444.16049
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.