Ilaria Colazzo, I.Colazzo (at)
University of Exeter (UK)
Paul Truman, p.j.truman (at)
Keele University (UK)
Leandro Vendramin, Leandro.Vendramin (at)
Vrije Universiteit Brussel (Belgium)

In 2016 a connection emerged between two apparently disparate topics: Hopf–Galois theory (whose applications include generalizations of the classical Galois correspondence and questions concerning the structure of rings of algebraic integers) and the theory of skew braces (intensively studied due to their connection with solutions of the Yang-Baxter equation, which appears in statistical mechanics, knot theory, …). This connection has already proven to be very fruitful, with numerous results concerning the existence, classification, and properties of objects being translated between topics, or developed in tandem.

The main aim of these meetings is to explore this interplay between skew braces and Hopf–Galois structures. Each side has its own concrete problems, with possible applications to understanding the mathematical structures belonging to the other side.

The structure of skew braces

When: 07 – 08 September 2023

Where: University of Exeter

07th September

  • Eric Jespers: Algebraic structure of Yang-Baxter algebras. (14:30-15:30)
  • Tea break (15:30-15:50)
  • Paul Truman: Skew bracoids and the Yang–Baxter equation. (15:50-16:20)
  • Isabel Martin-Lyons: Skew Bracoids in the Holomorph. (16:20-16:50)
  • Joao Faria Martins: Homotopy-theoretical constructions of biracks, welded-biracks, and generalisations. (16:50-17:50)

08th September

  • Arne Van Antwerpen: The non-fixing graph of a skew brace. (09:00-10:00)
  • Senne Trappenniers: Indecomposable solutions of the Yang–Baxter equation, generators of skew braces and Hopf–Galois structures. (10:00-10:30)
  • Coffee break (10:30-10:50)
  • Nigel Byott: A Family of Simple Skew Braces. (10:50–11:50)
  • Ilaria Colazzo: Cabling for non-involutive solutions. (11:50-12:20)

Hopf–Galois structures

When: 18 April 2023

Where: Keele University

  • Leandro Vendramin: Some problems on skew braces. (09:00-09:30)
  • Arne Van Antwerpen: Nilpotency in skew braces: an overview. (09:30-10:30)
  • Coffee (10:30-11:00)
  • Silvia Properzi: A graph associated with finite skew braces. (11:00-12:00)
  • Ilaria Colazzo: The structure monoid of set-theoretic solutions to the YBE. (12:00-13:00)
  • Lunch (01:00-02:00)
  • Lorenzo Stefanello: Skew braces and the Hopf-Galois correspondence. (02:00-03:00)
  • Tea (02:30-03:00)
  • Paul Truman: Skew bracoids and Hopf-Galois structures on separable extensions. (03:30-04:30)