Skew left braces with non-trivial annihilator. (English) Zbl 1447.16035

Summary: We describe the class of all skew left braces with non-trivial annihilator through ideal extension of a skew left brace. The ideal extension of skew left braces is a generalization to the non-abelian case of the extension of left braces provided by D. Bachiller in [J. Pure Appl. Algebra 222, No. 7, 1670–1691 (2018; Zbl 1437.20031)].


16T25 Yang-Baxter equations
16N20 Jacobson radical, quasimultiplication
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
20B15 Primitive groups


Zbl 1437.20031
Full Text: DOI


[1] Bachiller, D., Extensions, matched products, and simple braces, J. Pure Appl. Algebra222 (2018) 1670-1691. · Zbl 1437.20031
[2] Catino, F., Colazzo, I. and Stefanelli, P., On regular subgroups of the affine group, Bull. Aust. Math. Soc.91 (2015) 76-85. · Zbl 1314.20001
[3] Catino, F., Colazzo, I. and Stefanelli, P., Semi-braces and the Yang-Baxter equation, J. Algebra483 (2017) 163-187. · Zbl 1385.16035
[4] Catino, F. and Rizzo, R., Regular subgroups of the affine group and radical circle algebras, Bull. Aust. Math. Soc.79 (2009) 103-107. · Zbl 1184.20001
[5] Cedó, F., Jespers, E. and Okniński, J., Braces and the Yang-Baxter equation, Comm. Math. Phys.327 (2014) 101-116. · Zbl 1287.81062
[6] Drinfel’d, V. G., On some unsolved problems in quantum group theory, Quantum Groups (Leningrad, 1990), , Vol. 1510 (Springer, Berlin, 1992) pp. 1-8. · Zbl 0765.17014
[7] Etingof, P., Schedler, T. and Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J.100 (1999) 169-209. · Zbl 0969.81030
[8] Gateva-Ivanova, T. and Van den Bergh, M., Semigroups of \(I\)-type, J. Algebra206 (1998) 97-112. · Zbl 0944.20049
[9] Guarnieri, L. and Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comp.86 (2017) 2519-2534. · Zbl 1371.16037
[10] Lu, J.-H., Yan, M. and Zhu, Y.-C., On the set-theoretical Yang-Baxter equation, Duke Math. J.104 (2000) 1-18. · Zbl 0960.16043
[11] Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra307 (2007) 153-170. · Zbl 1115.16022
[12] Rump, W., The brace of a classical group, Note Mat.34 (2014) 115-144. · Zbl 1344.14029
[13] Soloviev, A., Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation, Math. Res. Lett.7 (2000) 577-596. · Zbl 1046.81054
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.