Related papers: Braided Hopf Crossed Modules Through Simplicial St…

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three…

We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. Indeed, any monoidal adjunction between autonomous…

We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and…

In this dissertation we study the coefficients spaces (SAYD modules) of Hopf-cyclic cohomology theory over a certain family of bicrossed product Hopf algebras, and we compute the Hopf-cyclic cohomology of such Hopf algebras with…

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's…

The aim of this paper is to construct a new braided $T$-category via the generalized Yetter-Drinfel'd modules and Drinfel'd codouble over Hopf algebra, an approach different from that proposed by Panaite and Staic \cite{PS}. Moreover, in…

This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras $A$ by first determining the graded Hopf algebra $\gr A$ associated to the coradical filtration of…

Within the framework of braided or quasisymmetric monoidal categories braided Q-supersymmetry is investigated, where Q is a certain functorial isomorphism in a braided symmetric monoidal category. For an ordinary (co-)quasitriangular Hopf…

Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of…

In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a…

Let $H$ be a finite dimensional semisimple Hopf algebra and $R$ a braided Hopf algebra in the category of Yetter-Drinfeld modules over $H$. When $R$ is a Calabi-Yau algebra, a necessary and sufficient condition for $R#H$ to be a Calabi-Yau…

We present characterizations of braided co-Frobenius Hopf algebras in the braided tensor category of Yetter-Drinfeld modules over a Hopf algebra extending those already known for co-Frobenius Hopf algebras.

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 $E$…

Let $\left( H,R\right) $ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${}…

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…

Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd…

Let $(H,\a_H)$ be a Hom-Hopf algebra, $(A,\a_A)$ a right $H$-comodule algebra and $(C,\a_C)$ a left $H$-module coalgebra. Then we have the category $_A\mathcal{M}(H)^C$ of Hom-type Doi-Hopf modules. The aim of this paper is to make the…

We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the…

We prove a structure theorem for Yetter-Drinfel'd Hopf algebras over groups of prime order that are nontrivial, cocommutative, and cosemisimple: Under certain assumptions on the base field, these algebras can be decomposed into a tensor…

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.…