@article { ,
title = {Opposite Skew Left Braces and Applications},
abstract = {Given a skew left brace B, we introduce the notion of an \\opposite" skew left brace B0, which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by B0 is the inverse to the solution given by B. Every Hopf-Galois structure on a Galois field extension L=K gives rise to a skew left brace B; if the underlying Hopf algebra is not commutative, then one can construct an additional, \\commuting" Hopf-Galois structure (see [10], which relates the Hopf-Galois module structures of each); the corresponding skew left brace to this second structure is precisely B0. We show how left ideals (and a newly introduced family of quasi-ideals) of B0 allow us to identify the intermediate fields of L=K which occur as fixed fields of sub-Hopf algebras under this correspondence. Finally, we use the opposite to connect the inverse solution to the YBE and the structure of the Hopf algebra H acting on L=K; this allows us to identify the group-like elements of H.},
doi = {10.1016/j.jalgebra.2019.10.033},
issn = {0021-8693},
journal = {Journal of Algebra},
note = {This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Elsevier at https://doi.org/10.1016/j.jalgebra.2019.10.033 - please refer to any applicable terms of use of the publisher.},
pages = {218-235},
publicationstatus = {Published},
publisher = {Elsevier},
keyword = {QA Mathematics, Skew left braces, Hopf-Galois structure, Yang-Baxter equation},
year = {2020},
author = {Koch, Alan and Truman, Paul}
}