Opposite skew left braces and applications

• Published in: Journal of algebra Vol. 546; pp. 218 - 235
• Main Authors: Koch, Alan, Truman, Paul J.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.03.2020
• Subjects: Hopf-Galois structure; Skew left braces; Yang-Baxter equation; 16T05; Yang-Baxter equation; Skew left braces; 16T25; 20N99; Hopf-Galois structure
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2019.10.033

•Introduces the notion of an opposite to a (skew left) brace.•(Known) Skew left braces provide solutions to the Yang-Baxter equation; (new) opposite braces provide inverse solutions.•The inverse solution to the YBE gives information about the coalgebra structure of a Hopf-Galois structure.•The opposite brace gives information on the intermediate fields found through the associated Hopf-Galois correspondence. Given a skew left brace B, we introduce the notion of an “opposite” skew left brace B′, 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 B′ 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 “opposite” Hopf-Galois structure (see [1], which relates the Hopf-Galois module structures of each, and refers to the structures as “commuting”); the corresponding skew left brace to this second structure is precisely B′. We show how left ideals (and a newly introduced family of quasi-ideals) of B′ allow us to identify the intermediate fields of L/K which occur as fixed fields of sub-Hopf algebras under this correspondence and to identify which of these are Galois, or Hopf-Galois, over K. 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.