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; this allows us to identify the group-like elements in the Hopf algebra providing the Hopf-Galois structure using only these solutions. We also show how left ideals of B ′ correspond to the realizable intermediate fields of a certain Hopf-Galois extension of a Galois extension.

Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

Let L/K be a finite separable extension of fields whose Galois closure E/K has group G. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L/K has the form E[N ] G for some group N such that |N | = [L : K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and Kalgebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree p n , for p an odd prime number.

Hopf-Galois theory was first described by S. U. Chase and M. E. Sweedler [Hopf algebras and Galois theory. Springer, Cham (1969; Zbl 0197.01403)], with the hope that it might shed light on the study of inseparable extensions. However, the theory applies for separable extensions as well, in which case there is a group-theoretic classification of Hopf-Galois structures due to C. Greither and B. Pareigis [J. Algebra 106, 239-258 (1987; Zbl 0615.12026)]. In the setting of Galois extensions, which is the main focus of the book under review, there is a one-to-one correspondence between the Hopf-Galois structures on a Galois G-extension and the regular subgroups normalized by left translations in the symmetric group of G. The type of a Hopf-Galois structure is defined to be the isomorphism class of the associated regular subgroup.Classical Galois module theory is about the study of the Galois module structure of rings of integers in number fields. A famous theorem of Leopoldt states that if L/Q is any abelian G-extension, then the ring of integers O L of L is free over the associated order. However, this does not generalize, even at the local level, when one considers relative or non-abelian extensions. N. P. Byott [in: Algebraic number theory and Diophantine analysis. Proceedings of the international conference, Graz, Austria, 1998. Berlin: Walter de Gruyter. 55-67 (2000; Zbl 0958.11076)] obtained examples of local Galois G-extensions L/K for which O L is not free over the classical associated orderbut is free over the associated order A H of L in some other Hopf-Galois structure H. This interesting result suggests that the consideration of various Hopf-Galois structures, other than the classical Hopf-Galois structure K[G], can broaden the scope of Galois module theory.

Given a finite group [Formula: see text], we study certain regular subgroups of the group of permutations of [Formula: see text], which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to [Formula: see text] and Hopf–Galois structures admitted by a Galois extension of fields with Galois group isomorphic to [Formula: see text]. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf–Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf–Galois structure is determined by the corresponding skew left brace. We investigate these questions in the context of a variety of existing constructions in the literature. As an application of our results we classify the isomorphically distinct Hopf algebras that give Hopf–Galois structures on a Galois extension of degree [Formula: see text] for [Formula: see text] prime numbers.