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Let $H$ be a semisimple Hopf algebra, and let $R$ be a noetherian left $H$-module algebra. If $R/R^H$ is a right $H^*$-dense Galois extension, then the invariant subalgebra $R^H$ will inherit the AS-Cohen-Macaulay property from $R$ under some mild conditions, and $R$, when viewed as a right $R^H$-module, is a Cohen-Macaulay module. In particular, we show that if $R$ is a noetherian complete semilocal algebra which is AS-regular of global dimension 2 and $H=operatorname{bf k} G$ for some finite subgroup $Gsubseteq Aut(R)$, then all the indecomposable Cohen-Macaulay module of $R^H$ is a direct summand of $R_{R^H}$, and hence $R^H$ is Cohen-Macaulay-finite, which generalizes a classical result for commutative rings. The main tool used in the paper is the extension groups of objects in the corresponding quotient categories.
We study invariants and quotient categories of fixed subrings of Artin-Schelter regular algebras under Hopf algebra actions.
We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from
The notions of Galois and cleft extensions are generalized for coquasi-Hopf algebras. It is shown that such an extension over a coquasi-Hopf algebra is cleft if and only if it is Galois and has the normal basis property. A Schneider type theorem is p
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 o
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