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Register a new userRepresentations of Hopf-Ore extensions of group algebras
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In this paper, we study the representations of the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(chi^{-1}, a, 0)$-modules under the assumption $|chi|=infty$ and $|chi|=|chi(a)|<infty$ respectively, and all finite dimensional indecomposable $kG(chi^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|chi|=|chi(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(chi^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra $kD_n$, where $n=2m$, $m>1$ odd, and char$(k)$=0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.
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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.
In this paper we study the theory of cleft extensions for a weak bialgebra H. Among other results, we determine when two unitary crossed products of an algebra A by H are equivalent and we prove that if H is a weak Hopf algebra, then the categories of H-cleft extensions of an algebra A, and of unitary crossed products of A by H, are equivalent.
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.
Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.
We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.
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