Hopf-Galois Theory and Skew Braces

The project will explore and develop the connection between two rather different areas of algebra, namely Hopf-Galois theory and (skew) braces.

Galois theory is the area of algebra which studies the symmetries (automorphisms) of number systems (fields) using groups. A familiar example is complex conjugation, which is an automorphism of the complex numbers that fixes all real numbers and generates a group of order 2, the Galois group of the complex numbers as an extension of the real numbers. Among other things, Galois theory explains why a polynomial equation of degree 5 or more cannot in general be solved by a formula. Hopf-Galois theory generalises this classical situation by replacing the Galois group with a Hopf algebra. A given field extension may have many Hopf-Galois structures, and, by a result of Greither and Pareigis (1987), finding them all amounts to a combinatorial problem in group theory. There are many results enumerating or restricting Hopf-Galois structures for extensions with various Galois groups, and the Principal Investigator has been a key contributor to this endeavour.

(Skew) braces are algebraic objects which give solutions of the Yang-Baxter equation. They are of considerable interest since the Yang-Baxter equation plays a fundamental role in many areas of theoretical physics and of mathematics. Braces were introduced by Rump (2007) and have since been studied by many authors. They were generalised to skew braces by Guarnieri and Vendramin (2017). It was first noted by Bachiller (2016) that there is a connection between Hopf-Galois structures and braces. This connection in fact extends to skew braces. The connection comes about because both Hopf-Galois structures and skew braces correspond to regular subgroups in the holomorph of another group. This means that there is a correspondence between Hopf-Galois structures and skew braces which, while not one-to-one, allows results on Hopf-Galois structures to be reinterpreted as results on skew braces, and vice versa.

This project will investigate several important open problems on braces and skew braces, and their analogues for Hopf-Galois structures. These problems all involve the notion of extensions of braces or Hopf-Galois structures, and the initial phase of the research will be to understand these extensions from a variety of perspectives. The insights gained from doing so will be then be used to enumerate quaternionic and dihedral braces, to study skew braces with insoluble multiplicative group and soluble additive group, to look for new examples of soluble groups which are not involutive Yang-Baxter groups, and to classify some classes of simple braces and simple skew braces.

The proposed research will have a significant impact on our understanding both of Hopf-Galois theory and of skew braces, and (via tha latter) can be expected to lead to constructions of new set-theoretic solutions of the Yang-Baxter Equation. Both the techniques developed, and the results proved, will inform future research in these areas and suggest new directions for investigation. As well as contributing to these specific areas of algebra, it will have implications for related areas of mathematics and physics, such as number theory, knot theory and statistical mechanics.

This project will build bridges between the two communities of researchers in Hopf-Galois structures and in algebraic approaches to the Yang-Baxter equation. It will therefore strengthen the international links of U.K. research in algebra, particularly with European and Argentinian mathematicians. These will be fostered in several ways. The project will involve visits between the PI/PDRA and leading contributors to the theory of braces in Spain and Argentina. The PI and PDRA will present their results at national and international conferences, workshops and seminars. They will also organise a workshop in Exeter towards the end of the project, bringing together researchers from both communities. Results obtained, once written up, will be circulated as preprints within both communities, and posted on arXiv as soon as practicable. We expect new collaborations and research directions to result from this.

The project will enable the employment and career development of the PDRA, who by the end of the project will have established her/himself as an active research mathematician with strong publication record, and who will be well-placed to find a permanent academic post.

As with much research in pure mathematics, specific impact outside the academic realm is difficult to predict. The new algebraic insights and specific constructions arising from the project may in the longer term have applications in areas such as information security, in particular coding theory or cryptography.