Keele Research Repository

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.