- Date
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Thursday 19 December 2024, 10.00 - 11.00
- Abstract
- Chase and Sweedler showed that Hopf algebras can be used to generalise many of the ideas of Galois theory to field extensions that are inseparable or non-normal; their techniques also have applications to extensions that are already Galois in the classical sense. Work of Greither and Pareigis and of Byott showed that the so-called Hopf-Galois structures on a given Galois extension of fields can be classified in terms of regular subgroups of holomorphs of groups. When Guarnieri and Vendramin introduced skew braces (building on Rump's definition of braces), it was found that these objects can also be classified via regular subgroups of holomorphs, and so skew brace theory is connected with Hopf-Galois theory. We will survey the fruitful interplay between these topics, showing how existence, classification, and structural results have been translated between the two languages, and how further generalizations of skew braces such as bracoids (largely motivated by questions arising in Hopf-Galois theory) and semibraces (designed to study degenerate solutions of the set theoretic Yang-Baxter equation) have turned out to be closely related to one another.
