On derived-indecomposable solutions of the Yang-Baxter equation

• Autores: ILARIA COLAZZO, M. Ferrara, Marco Trombetti
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 69, Nº 1, 2025, págs. 171-193
• Idioma: inglés
• DOI: 10.5565/publmat6912508
• Enlaces
  • Texto completo
• Resumen

  • If (X, r) is a finite non-degenerate set-theoretic solution of the Yang-Baxter equation, the additive group of the structure skew brace G(X, r) is an F C-group, i. e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to being an F C-group itself. If one additionally assumes that the derived solution of (X, r) is indecomposable, then for every element b of G(X, r) there are finitely many elements of the form b∗c and c ∗ b, with c ∈ G(X, r). This naturally leads to the study of a brace-theoretic analogue of the class of F C-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories, and that they behave well with respect to certain nilpotency concepts and finite generation.

• Referencias bibliográficas
  • E. Acri and M. Bonatto, Skew braces of size pq, Comm. Algebra 48(5) (2020), 1872–1881. DOI: 10.1080/00927872.2019.1709480
  • D. Bachiller, Classification of braces of order p3, J. Pure Appl. Algebra 219(8) (2015), 3568–3603. DOI: 10.1016/j.jpaa.2014.12.013
  • R. Baer, Endlichkeitskriterien f¨ur Kommutatorgruppen, Math. Ann. 124 (1951), 161–177. DOI: 10.1007/BF01343558
  • R. J. Baxter, Partition function of the Eight-Vertex lattice model, Ann. Physics 70(1) (1972), 193–228. DOI: 10.1016/0003-4916(72)90335-1
  • M. Bonatto and P. Jedlicka ˇ , Central nilpotency of skew braces, J. Algebra Appl. 22(12) (2023), Paper no. 2350255, 16 pp. DOI: 10.1142/S0219498823502559
  • S. Camp-Mora and R. Sastriques, A criterion for decomposability in QYBE, Int. Math. Res. Not. IMRN 2023(5) (2023), 3808–3813. DOI: 10.1093/imrn/rnab357
  • M. Castelli, Classification of uniconnected involutive solutions of the Yang–Baxter equation with odd size and a Z-group permutation group,...
  • M. Castelli, M. Mazzotta, and P. Stefanelli, Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation, Forum Math....
  • M. Castelli, G. Pinto, and W. Rump, On the indecomposable involutive set-theoretic solutions of the Yang–Baxter equation of prime-power size,...
  • F. Catino, I. Colazzo, and P. Stefanelli, Skew left braces with non-trivial annihilator, J. Algebra Appl. 18(2) (2019), 1950033, 23 pp. DOI:...
  • F. Cedo, E. Jespers, L. Kubat, A. Van Antwerpen, and C. Verwimp ´ , On various types of nilpotency of the structure monoid and group of a...
  • F. Cedo, E. Jespers, and J. Okni ´ nski ´ , Braces and the Yang–Baxter equation, Comm. Math. Phys. 327(1) (2014), 101–116. DOI: 10.1007/s00220-014-1935-y
  • F. Cedo, A. Smoktunowicz, and L. Vendramin ´ , Skew left braces of nilpotent type, Proc. Lond. Math. Soc. (3) 118(6) (2019), 1367–1392. DOI:...
  • S. N. Cernikov ˇ , On the structure of groups with finite classes of conjugate elements (Russian), Dokl. Akad. Nauk SSSR (N.S.) 115 (1957),...
  • M. De Falco, F. de Giovanni, C. Musella, and Y. P. Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc. 139(2) (2011),...
  • V. G. Drinfel’d, On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer-Verlag,...
  • P. Etingof, T. Schedler, and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100(2) (1999), 169–209....
  • P. Etingof, A. Soloviev, and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang–Baxter equation on a set with a prime...
  • L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comp. 86(307) (2017), 2519–2534. DOI: 10.1090/mcom/3161
  • P. Jedlicka, A. Pilitowska, and A. Zamojska-Dzienio ˇ , Cocyclic braces and indecomposable cocyclic solutions of the Yang–Baxter equation,...
  • E. Jespers, L. Kubat, and A. Van Antwerpen, The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter...
  • E. Jespers, L. Kubat, A. Van Antwerpen, and L. Vendramin, Radical and weight of skew braces and their applications to structure groups of...
  • E. Jespers, A. Van Antwerpen, and L. Vendramin, Nilpotency of skew braces and multipermutation solutions of the Yang–Baxter equation, Commun....
  • V. Lebed, S. Ram´ırez, and L. Vendramin, Involutive Yang–Baxter: cabling, decomposability, and Dehornoy class, Rev. Mat. Iberoam. 40(2) (2024),...
  • V. Lebed and L. Vendramin, On structure groups of set-theoretic solutions to the Yang–Baxter equation, Proc. Edinb. Math. Soc. (2) 62(3) (2019),...
  • J.-H. Lu, M. Yan, and Y.-C. Zhu, On the set-theoretical Yang–Baxter equation, Duke Math. J. 104(1) (2000), 1–18. DOI: 10.1215/S0012-7094-00-10411-5
  • A. Mann, The exponents of central factor and commutator groups, J. Group Theory 10(4) (2007), 435–436. DOI: 10.1515/JGT.2007.035
  • M. M. Parmenter and C. Polcino Milies, Group rings whose units form a nilpotent orFC group, Proc. Amer. Math. Soc. 68(2) (1978), 247–248....
  • C. Polcino Milies, Group rings whose units form an FC-group, Arch. Math. (Basel) 30(4) (1978), 380–384. DOI: 10.1007/BF01226070
  • C. Polcino Milies, Group rings whose units form an FC-group: Corrigendum, Arch. Math. (Basel) 31(5) (1978), 528. DOI: 10.1007/BF01226485
  • S. Ram´ırez, Indecomposable solutions of the Yang–Baxter equation with permutation group of sizes pq and p2q, Comm. Algebra 51(10) (2023),...
  • S. Ram´ırez and L. Vendramin, Decomposition theorems for involutive solutions to the Yang–Baxter equation, Int. Math. Res. Not. IMRN 2022(22)...
  • W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193(1) (2005), 40–55. DOI:...
  • W. Rump, Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307(1) (2007), 153–170. DOI: 10.1016/j.jalgebra.2006.03.040
  • W. Rump, One-generator braces and indecomposable set-theoretic solutions to the Yang–Baxter equation, Proc. Edinb. Math. Soc. (2) 63(3) (2020),...
  • W. Rump, Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups, Canad. Math. Bull. 65(1) (2022), 225–233. DOI:...
  • S. K. Sehgal and H. J. Zassenhaus, Group rings whose units form an FC-group, Math. Z. 153(1) (1977), 29–35. DOI: 10.1007/BF01214731
  • A. Smoktunowicz and A. Smoktunowicz, Set-theoretic solutions of the Yang–Baxter equation and new classes of R-matrices, Linear Algebra Appl....
  • A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra 2(1) (2018), 47–86. DOI:...
  • A. Soloviev, Non-unitary set-theoretical solutions to the quantum Yang–Baxter equation. Math. Res. Lett. 7(5-6) (2000), 577–596. DOI: 10.4310/MRL.2000.v7.n5.a4
  • M. J. Tomkinson, FC-groups, Res. Notes in Math. 96, Pitman (Advanced Publishing Program), Boston, MA, 1984.
  • C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967),...