Related papers: Hopf-Galois objects and cogroupoids

We give an account of the current state of the approch to quantum field theory via Hopf algebras and Hochschild cohomology. We emphasize the versatility and mathematical foundation of this algebraic structure, and collect algebraic…

We investigate the first two Galois cohomology groups of $p$-extensions over a base field which does not necessarily contain a primitive $p$th root of unity. We use twisted coefficients in a systematic way. We describe field extensions…

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups,…

We introduce the concept of a covering of a graded pointed Hopf algebra. The theory developed shows that coverings of a bosonized Nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering…

We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of…

Using methods from math-ph/9907010, we study families of Hopf algebra structures on coloured trees.

In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group $G$ is an almost simple group in which the socle $A$ has prime index $p$. Each Hopf-Galois structure is associated to a group $N$ of the same…

By work of C. Greither and B. Pareigis as well as N. P. Byott, the enumeration of Hopf-Galois structures on a Galois extension of fields with Galois group $G$ may be reduced to that of regular subgroups of $\mbox{Hol}(N)$ isomorphic to $G$…

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\Gamma$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded…

We describe the Hopf algebra quotients and Hopf images of the smash coproduct of a group algebra by the algebra of functions on a finite group.

For a finite-index $\mathrm{II}_1$ subfactor $N \subset M$, we prove the existence of a universal Hopf $\ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$…

In these lectures I discuss Hopf algebras and Dyson-Schwinger equations. The lectures start with an introduction to Hopf algebras, followed by a review where Hopf algebras occur in particles physics. The final part of these lectures is…

This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras…

We investigate local-global principles for Galois cohomology, in the context of function fields of curves over semi-global fields. This extends work of Kato's on the case of function fields of curves over global fields.

We construct cohomology theories for $(\varphi, \tau)$-modules, and study their relation with cohomology of $(\varphi, \Gamma)$-modules, as well as Galois cohomology. Our method is axiomatic, and can treat the \'etale case, the…

Given a finite group $ G $, we study certain regular subgroups of the group of permutations of $ G $, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to $ G…

Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The…

This paper uses Galois maps to give a definition of generalized multiplier Hopf coquasigroups, and give a sufficient and necessary condition for a multiplier bialgebra to be a regular multiplier Hopf coquasigroup. Then coactions and…

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation…

We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson--Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free…