Some group-theoretical approaches to skew left braces

Document Type : Ischia Group Theory 2020/2021


1 Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, Burjassot, València, Spain

2 Departamento de Matemáticas, Universidad de Zaragoza, Pedro Cerbuna, 12, Zaragoza, Spain


The algebraic structure of skew left brace has become a useful tool to construct set-theoretic solutions of the Yang-Baxter equation. In this survey we present some descriptions of skew left braces in terms of bijective derivations, triply factorised groups, and regular subgroups of the holomorph of a group, as well as some applications of these descriptions to the study of substructures, nilpotency, and factorised skew left braces.


Main Subjects

[1] E. Acri, R. Lutowski and L. Vendramin, Rectractability of solutions to the Yang-Baxter equation and p-nilpotency
of skew braces, Internat. J. Algebra Comput., 30 no. 1 (2020) 91–115.
[2] B. Amberg, S. Franciosi and F. de Giovanni, Products of groups, Oxford Mathematical Monographs, The Clarendon
Press Oxford University Press, New York, Oxford Science Publications, 1992.
[3] D. Bachiller, F. Cedó and E. Jespers, Solutions of the Yang-Baxter equation associated with a left brace, J. Algebra,
463 (2016) 80–102.
[4] A. Ballester-Bolinches and R. Esteban-Romero, Triply factorised groups and the structure of skew left braces,
Commun. Math. Stat., (in press).
[5] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of finite groups, de Gruyter Expositions in
Mathematics, 53, Walter de Gruyter GmbH & Co. KG, Berlin, 2010.
[6] A. Ballester-Bolinches, R. Esteban-Romero, P. Jiménez-Seral and V. Pérez-Calabuig, On Yang-Baxter groups,
[7] R. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I, Some funda-
mental eigenvectors, Ann. Physics, 76 no. 1 (1973) 1–24.
[8] F. Cedó, E. Jespers and Á. del Rı́o, Involutive Yang-Baxter groups, Trans. Amer. Math. Soc., 362 no. 5 (2010)
[9] K. Doerk and T. Hawkes, Finite soluble groups, De Gruyter Expositions in Mathematics, 4, Walter de Gruyter &
Co., Berlin, 1992.
[10] V. G. Drinfeld, On some unsolved problems in quantum group theory, Quantum groups (Leningrad, 1990), Lecture
Notes in Math., 1510, Springer, Berlin, 1992 1–8.
[11] F. Eisele, On the IYB-property in some solvable groups, Arch. Math. (Basel), 101 (2013) no. 4 309–318.
[12] P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math.
J., 100 (1999) 169–209.
[13] L. Guarnieri and L. Vendramin, Skew-braces and the Yang-Baxter equation, Math. Comp., 86 no. 307 (2017)
[14] E. Jespers, L. Kubat, A. Van Antwerpen and L. Vendramin, Factorizations of skew braces, Math. Ann., 375 no. 3-4
(2019) 1649–1663.
[15] H. Meng, A. Ballester-Bolinches and R. Esteban-Romero, Left braces and the quantum Yang-Baxter equation, Proc.
Edinburgh Math. Soc., 62 no. 2 (2019) 595–608.
[16] H. Meng, A. Ballester-Bolinches, R. Esteban-Romero and N. Fuster-Corral, On finite involutive Yang–Baxter groups,
Proc. Amer. Math. Soc. 149 no. 2 (2021) 793–804.
[17] W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307 (2007) 153–170.
[18] A. Smoktunowicz, On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation,Trans. Amer.
Math. Soc., 370 no. 9 (2018) 6535–6564.
[19] A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb.
Algebra, 2 no. 1 (2018) 47–86.
[20] Y. P. Sysak, Products of groups and quantum Yang-Baxter equation, Notes of a talk in Advances in Group Theory
and Applications, Porto Cesareo, Lecce, Italy, 2011.
[21] C. N. Yang, Some exact results for many-body problem in one dimension with repulsive delta-function interaction,
Phys. Rev. Lett, 19 (1967) 1312–1315.
Volume 12, Issue 2 - Serial Number 2
Proceedings of the Ischia Group Theory (2020/2021) - Part 2
June 2023
Pages 99-109
  • Receive Date: 07 January 2022
  • Revise Date: 27 April 2022
  • Accept Date: 09 March 2022
  • Published Online: 01 June 2023