Related papers: Modules over invertible 1-cocycles

Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of…

Let $H$ be a Hopf algebra in braided category $\cal C$. Crossed modules over $H$ are objects with both module and comodule structures satisfying some comatibility condition. Category ${\cal C}^H_H$ of crossed modules is braided and is…

In this paper, we introduce the notion of a four-angle Hopf module for a Hom-Hopf algebra $(H,\beta)$ and show that the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ of four-angle Hopf modules is a monoidal category with either a Hom-tensor…

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 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…

For each braided category $\mathcal{C}$ we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to $\mathcal{C}$ which is not only monoidal but even braided and balanced.…

Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In…

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…

Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a…

We consider Hopf bimodules and crossed modules over a Hopf algebra $H$ in a braided category. They are the key-stones for braided bicovariant differential calculi and their invariant vector fields respectively, as well as for the…

We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the…

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be…

Starting with a self-dual Hopf algebra H in a braided monoidal category S we construct a Z/2Z-graded monoidal category C = C_0 + C_1. The degree zero component is the category Rep_S(H) of representations of H and the degree one component is…

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space ${\mathfrak A}\ot H^*$ we can define an…

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.

We classify exact indecomposable module categories over the representation category of all non-trivial Hopf algebras with coradical S_3 and S_4. As a byproduct, we compute all its Hopf-Galois extensions and we show that these Hopf algebras…

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category $\mathcal{C}$ endowed with a symmetric $2$-trace, one can attach a cyclic (resp. cocyclic)…

Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce all possible categories of Yetter-Drinfeld modules over $H$, and give explicitly the monoidal and braided structure of them. Then we prove that the category…

This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid…

Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of…