Opposite skew left braces and applications

Abstract

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.