2000 character limit reached
Indecomposable involutive set-theoretical solutions to the Yang-Baxter equation of size
Published 27 Mar 2024 in math.QA, math.GR, and math.RA | (2403.18653v3)
Abstract: The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions to the Yang--Baxter equation, whose investigation is strongly driven by the study of algebraic objects called (skew) braces. In this article, we focus on indecomposable involutive non-degenerate set-theoretic solutions to the Yang-Baxter equation. More specifically, through a thorough analysis of their associated braces, we give a full classification of those which are of size , for a prime.
- Enumeration of set-theoretic solutions to the Yang-Baxter equation. Math. Comp., 91:1469–1481, 2020.
- A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math., 29(6):1291–1306, 2017.
- R. J. Baxter. Partition function of the eight-vertex lattice model. Ann. Physics, 70(1):193–228, 1972.
- M. Castelli and S. Trappeniers. Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group. arXiv.2303.00581, 2023.
- Braces and the Yang-Baxter equation. Comm. Math. Phys., 327(1):101–116, 2014.
- F. Cedó and J. Okniński. Constructing finite simple solutions of the Yang-Baxter equation. Adv. Math., 391:Paper No. 107968, 40, 2021.
- F. Cedó and J. Okniński. Indecomposable solutions of the Yang-Baxter equation of square-free cardinality. Adv. Math., 430:Paper No. 109221, 26, 2023.
- V. G. Drinfeld. On some unsolved problems in quantum group theory. In P. P. Kulish, editor, Quantum Groups, pages 1–8, Berlin, Heidelberg, 1992. Springer Berlin Heidelberg.
- Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Math. J., 100(2):169 – 209, 1999.
- E. Feingesicht. Dehornoy’s class and Sylows for set-theoretical solutions of the Yang-Baxter equation. Internat. J. Algebra Comput., 34(1):147–173, 2024.
- T. Gateva-Ivanova and M. Van den Bergh. Semigroups of I-type. J. Algebra, 206:97–112, 1998.
- L. Guarnieri and L. Vendramin. Skew braces and the Yang-Baxter equation. Math. Comp., 86(307):pp. 2519–2534, 2017.
- B. Huppert. Endliche Gruppen. Finite groups / B. Huppert, N. Blackburn. Springer, 1983.
- P. Jedlička and A. Pilitowska. Indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation group. Journal of Combinatorial Theory, Series A, 197:105753, 2023.
- Involutive Yang-Baxter: cabling, decomposability, and Dehornoy class. Rev. Mat. Iberoam., 40(2):623–635, 2024.
- A. Lucchini. On imprimitive groups with small degree. Rend. Semin. Mat. Univ. Padova, 86:131–142, 1991.
- W. Rump. A decomposition theorem for square-free unitary solutions of the quantum yang-baxter equation. Adv. Math., 193(1):40–55, 2005.
- W. Rump. Braces, radical rings, and the quantum Yang–Baxter equation. J. Algebra, 307(1):153–170, 2007.
- W. Rump. Primes in coverings of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation. Bull. Belg. Math. Soc. Simon Stevin, 30(2):260–280, 2023.
- A. Smoktunowicz and L. Vendramin. On skew braces (with an appendix by N. Byott and L. Vendramin). J. Comb. Algebra, 2(1):47–86, 2018.
- V. G. Turaev. The Yang-Baxter equation and invariants of links. Invent. Math., 92:527–553, 1988.
- C. N. Yang. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett., 19:1312–1315, Dec 1967.
Paper Prompts
Sign up for free to create and run prompts on this paper.