nLab
skew brace
Context
Algebra
- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law
Group theory
Ring theory
Module theory
Contents
Idea
A skew brace is a set equipped with a pair of group structures satisfying some compatibility conditions.
This is a generalization of the concept of brace (an algebraic structure due W. Rump), which it reduces to when one of the groups is Abelian.
Skew baces can be used to construct solutions to the set-theoretic Yang-Baxter equation; their actions can be used to construct solutions of the reflection equation.
Literature
Introduced in
- L. Guarnieri, L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comp. 86 (2017) 2519–2534 doi
Viewpoint via sup-lattices
- Aryan Ghobadi, Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat, J. Algebra 586 (2021) 607–642 doi
In construction of solutions to set-theoretic Yang–Baxter equation,
- Valeriy G. Bardakov, Vsevolod Gubarev, Rota–Baxter groups, skew left braces, and the Yang–Baxter equation, Journal of Algebra 596 (2022) 328–351 doi
In construction of solutions of the reflection equation
- Kenny De Commer, Actions of skew braces and set-theoretic solutions of the reflection equation, Proc. Edinburgh Math. Soc. 62:4 (2019) 1089–1113 doi
Last revised on October 1, 2024 at 11:32:07.
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