ﻻ يوجد ملخص باللغة العربية
Let $left( H,Rright) $ 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 ${} {}_{H}^{H}mathcal{YD}.$ Let $H_{R}$ be the Majids transmuted braided group of $left( H,Rright) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}opluscdotsoplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $left( 1leq ileq rright) $, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define the $R$-adjoint-stable algebra $N_{W_{i}}$ of $W_{i}$. Using Ostriks theorem on characterizing module categories over monoidal categories, we prove that $Vin{}_{H}^{H}mathcal{YD}$ is irreducible if and only if there exists an $i$ $left( 1leq ileq rright) $ and an irreducible right $N_{W_{i}}$-module $U_{i}$, such that $Vcong U_{i}otimes_{N_{W_{i}}}left( Hotimes W_{i}right) $. Our structure theorem generalizes the results of Dijkgraaf-Pasquier-Roche and Gould on Yetter-Drinfeld modules over finite group algebras. If $k$ is an algebraically closed field of characteristic, we stress that the $R$-adjoint-stable algebra $N_{W_{i}}$ is an algebra over which the dimension of each irreducible right module divides its dimension.
Let $(R^{vee},R)$ be a dual pair of Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra $H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter-Drinfeld modules over the
Given a Hopf algebra $H$ and a projection $Hto A$ 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 techniques in th
If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of deformations that h
Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the
In this paper, we continue our study of the tensor product structure of category $mathcal W$ of weight modules over the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero