Representation theory of skew braces

Document Type : Research Paper

Authors

1 Department of Mathematics, Tokyo University of Science

2 Department of Mathematics, Ochanomizu University

Abstract

According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we shall explain how some of the results from representation theory of groups, such as Maschke's theorem and Clifford's theorem, extend naturally to that of skew braces. We shall also give some concrete examples to illustrate that skew brace representations are more difficult to classify than group representations.

Keywords

Main Subjects

References

[1] E. Acri and M. Bonatto, Skew braces of size pq, Comm. Algebra, 48 no. 5 (2020) 1872–1881.
[2] D. Bachiller, Classification of braces of order p3 , J. Pure Appl. Algebra, 219 no. 8 (2015) 3568–3603.
[3] F. CedoĢ, A. Smoktunowicz, and L. Vendramin, Skew left braces of nilpotent type, Proc. Lond. Math. Soc. (3), 118 no. 6 (2019) 1367–1392.
[4] L. Guarnieri and L. Vendramin, Skew braces and the Yang-Baxter equation, Math. Comp., 86 no. 307 (2017) 2519–2534.
[5] E. Jespers, L. Kubat, A. Van Antwerpen, and L. Vendramin, Factorizations of skew braces, Math. Ann., 375 no. 3-4 (2019) 1649–1663.
[6] E. Jespers, A. Van Antwerpen, and L. Vendramin, Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation, Commun. Contemp. Math., 25 no. 9 (2023) Paper No. 2250064, 20 pp.
[7] A. Koch and P. J. Truman, Opposite skew left braces and applications, J. Algebra, 546 (2020) 218–235.
[8] T. Letourmy and L. Vendramin, Isoclinism of skew braces, Bull. Lond. Math. Soc., 55 no. 6 (2023) 2891–2906.
[9] T. Letourmy and L. Vendramin, Schur covers of skew braces, J. Algebra, 644 (2024) 609–654.
[10] W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307 no. 1 (2007) 153–170.
[11] C. Tsang, A generalization of Ito’s theorem to skew braces, J. Algebra, 642 (2024) 367–399.
[12] P. Webb, A course in finite group representation theory, Cambridge Studies in Advanced Mathematics, 161 Cambridge University Press, Cambridge, 2016.
[13] H. Zhu, The construction of braided tensor categories from Hopf braces, Linear Multilinear Algebra, 70 no. 16 (2022) 3171–3188.
International Journal of Group Theory

Articles in Press, Accepted Manuscript
Available Online from 01 September 2024

Files

History

  • Receive Date: 27 July 2024
  • Revise Date: 27 August 2024
  • Accept Date: 01 September 2024
  • Published Online: 01 September 2024

Share

How to cite

Statistics