Related papers: Distributive laws and Hopf quasigroups

Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the…

A product system E over a semigroup P is a family of Hilbert spaces {E_s:s\in P} together with multiplications E_s \times E_t\to E_{st}. We view E as a unitary- valued cocycle on P, and consider twisted crossed products A \times_{\beta,E} P…

We associate, in a functorial way, a monoidal bicategory $\mathsf{Span}| \mathcal V$ to any monoidal bicategory $\mathcal V$. Two examples of this construction are of particular interest: Hopf polyads (due to Brugui\`eres) can be seen as…

In this paper we introduce the category of strong Hopf modules for a weak Hopf quasigroup H in a braided monoidal category. We also prove that this category is equivalent to the category of right modules over the image of the target…

We introduce two coproducts on mixed graphs (that is to say graphs with both oriented and unoriented edges), the first one by separation of the vertices into two parts, and the second one given by contraction and extractions of subgraphs.…

Let $D(H)$ be the quantum double associated to a finite dimensional quasi-Hopf algebra $H$. In this note, we first generalize a result of Majid, stating that a finite dimensional Hopf algebra $H$ is quasitriangular if and only if there is a…

Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable…

We prove the graded braided commutativity of the Hochschild cohomology of $A$ with trivial coefficients, where $A$ is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra of an abelian group, under some…

In Homotopy Type Theory, few constructions have proved as troublesome as the smash product. While its definition is just as direct as in classical mathematics, one quickly realises that in order to define and reason about functions over…

We compute the mod-2 cohomology of the collection of all symmetric groups as a Hopf ring, where the second product is the transfer product of Strickland and Turner. We first give examples of related Hopf rings from invariant theory and…

We investigate a semigroup construction related to the two-sided wreath product. It encompasses a range of known constructions and gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element…

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category $\mathcal{K}$. In order to do that, we introduce the 2-category of relative monads…

The unified product was defined in \cite{am3} related to the restricted extending structure problem for Hopf algebras: a Hopf algebra $E$ factorizes through a Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1\in H$ if and only if $E$…

Definitions of actions of Hopf quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf quasigroups and coquasigroups, and solutions to Militaru's D-equation are…

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components…

With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra $H$ over a field $k$ we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal…

Let $G$ be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra $K[G]$ (equivalently, any bicrossed product between…

To a finite group G one can associate a tower of wreath products S_n[G]. It is well known that the graded direct sum of the Grothendieck groups of the categories of finite dimensional complex representations of these groups can be given the…

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra, and $(A,\beta)$ a Hom-algebra. In this paper we will introduce the crossed product $(A\#_{\sigma}H,\beta\otimes\alpha)$, which is a Hom-algebra. Then we will introduce the notions of cleft…

We introduce a more general version of the so-called L-R-smash product and study its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product). We also give an interpretation of the…