Related papers: Yetter-Drinfeld modules for weak Hom-Hopf algebras

We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra $H_0$ in the category of…

Let $H$ be a finite dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra $H\o H^*$ as monoidal category. Moreover…

In the first part we recall two famous sources of solutions to the Yang-Baxter equation -- R-matrices and Yetter-Drinfel$'$d (=YD) modules -- and an interpretation of the former as a particular case of the latter. We show that this result…

For a Hopf algebra B with bijective antipode, we show that the Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra over the Drinfeld double D(B). The braiding structure allows generalizing H(B^*) =…

We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be…

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided…

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas $(H,\alpha)$ and $(B,\beta)$ and associate it with two nonlinear equations. We first introduce the notion of an $(H,B)$-Hom-Long dimodule and…

We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid…

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…

For a regular multiplier Hopf algebra $A$, the Yetter-Drinfel'd module category ${}_{A}\mathcal{YD}^{A}$ is equivalent to the centre $Z({}_{A}\mathcal{M})$ of the unital left $A$-module category ${}_{A}\mathcal{M}$. Then we introduce the…

The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel'd modules over a Hopf algebra, from self-distributive structures, and from…

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant…

Let $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some…

For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an…

Let $\pi$ be a group. The aim of this paper is to construct the category of Yetter-Drinfeld modules over the quasi-Turaev group coalgebra $H=(\{H_\a\}_{\a\in\pi},\Delta,\varepsilon,S,\Phi)$, and prove that this category is isomorphic to the…

We study versions of the categories of Yetter-Drinfel'd modules over a Hopf algebra $H$ in a braided monoidal category $\C$. Contrarywise to Bespalov's approach, all our structures live in $\C$. This forces $H$ to be transparent or…

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of…

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…