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Register a new userGreen ring of the category of weight modules over the Hopf-Ore extensions of group algebras
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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. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $chi$ and $chi(a)$ are different. Then we describe the Green ring $r(mathcal W)$ of the tensor category $mathcal W$. It is shown that $r(mathcal W)$ is isomorphic to the polynomial algebra over the group ring $mathbb{Z}hat{G}$ in one variable when $|chi(a)|=|chi|=infty$, and that $r(mathcal W)$ is isomorphic to the quotient ring of the polynomial algebra over the group ring $mathbb{Z}hat{G}$ in two variables modulo a principle ideal when $|chi(a)|<|chi|=infty$. When $|chi(a)|le|chi|<infty$, $r(mathcal W)$ is isomorphic to the quotient ring of a skew group ring $mathbb{Z}[X]sharphat{G}$ modulo some ideal, where $mathbb{Z}[X]$ is a polynomial algebra over $mathbb{Z}$ in infinitely many variables.
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In this paper, we investigate the tensor structure of the category of finite dimensional weight modules over the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebras $kG$. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that $k$ is an algebraically closed field of characteristic zero, and the orders of $chi$ and $chi(a)$ are the same.
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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 $A$ 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 Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^circ$ in $R^A$.
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