Related papers: Weak Multiplier Hopf Algebras. Preliminaries, moti…

The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough $\mathcal E$-projective objects with respect to the class…

We show that with the appropriate choice of coproduct, the type B quasisymmetric functions form a Hopf algebra, and the recently introduced type B peak functions form a Hopf subalgebra.

We classify pointed $p^3$-dimensional Hopf algebras $H$ over any algebraically closed field $k$ of prime characteristic $p>0$. In particular, we focus on the cases when the group $G(H)$ of group-like elements is of order $p$ or $p^2$, that…

Action of finite-dimensional Hopf algebra $H$ on commutative $k-$algebra $A$ is considered. As a generalization of the well-known fact for finite groups S. Montgomery raised a problem in 1993 whether $A$ is integral over subalgebra of…

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with…

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by…

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed…

Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non locally PI,…

We define the concept of \emph{companion automorphism} of a Hopf algebra $H$ as an automorphism $\sigma:H \rightarrow H$: $\sigma^2=S^2$ --where $S$ denotes the antipode--. A Hopf algebra is said to be \emph{almost involutive} (AI) if it…

Let $A=\bigoplus_{p\in G}A_{p}$ be a multiplier Hopf $T$-coalgebra over a group $G$, in this paper we give the definition of the crossed left $A$-$G$-modules and show that the category of crossed left $A$-$G$-modules is a monoidal category.…

Let $H$ be a finite-dimensional weak Hopf algebra over a field $k$ and $A/B$ be a right faithfully flat weak $H$-Galois extension. We prove that if the finitistic dimension of $B$ is finite, then it is less than or equal to that of $A$.…

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\text{adfin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely…

We show that, in order to classify Hopf algebra (co)actions on a given finite dimensional algebra up to equivalence, one should start with the classification of the possible cosupports (i.e. the sets of linear operators by which $H^*$ is…

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we…

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the…

We propose an algebraic study of the simple graph isomorphism problem. We define a Hopf algebra from an explicit realization of its elements as formal power series. We show that these series can be evaluated on graphs and count occurrences…

We investigate the splitting property of quasitriangular Hopf algebras through the lens of twisted tensor products. Specifically, we demonstrate that an infinite-dimensional quasitriangular Hopf algebra possesses the splitting property if…

The incidence algebra of a partially ordered set (poset) supports in a natural way also a coalgebra structure, so that it becomes a m-weak bialgebra even a m-weak Hopf algebra with M\"obius function as antipode. Here m-weak means that…

Let $A$ be a finite-dimensional Hopf algebra. The left and the right integrals on $A$ are related by means of a distinguished group-like element $\delta$ of $A$. Similarly, there is this element $\hat\delta$ in the dual Hopf algebra $\hat…

We use Hopf algebroids to formulate a notion of a noncommutative and non-cocommutative Hopf 2-algebra. We show how these arise from a bicrossproduct Hopf algebra with Peiffer identities. In particular, we show that for a Hopf algebra $H$…