Frobenius and separable functors for the category of entwined modules over cowreaths. II: applications
type: Item
instance of: scholarly article
title: Frobenius and separable functors for the category of entwined modules over cowreaths. II: applications
zbMATH Open document ID: 1472.16028
published in: Journal of Algebra
publication date: 2018-10-15T00:00:00Z
review text: Throughout the review, \(k\) denotes a field, \(H\) denotes a quasi-Hopf \(k\)-algebra, \(A\) denotes a right \(H\)-comodule \(k\)-algebra with \(\rho\colon A\rightarrow A\otimes H\), \(\rho (a)=\sum a_{\langle 0\rangle}\otimes a_{\langle 1\rangle}\), \(C\) denotes a right \(H\)-module \(k\)-coalgebra, \(\mathcal{M}_k\) denotes the category of \(k\)-vector spaces, \(\mathcal{M}_A\) denotes the category of right \(A\)-modules, \(\mathcal{M}(H)^C_A\) denotes the category of \((H,A,C)\)-Doi-Hopf modules. The paper under review is intended as the study of the Frobenius and separable properties for the forgetful functor \(F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A\). For \(\psi\colon C\otimes A\rightarrow A\otimes C\), \(\psi (c\otimes a)=\sum a_{\langle 0\rangle}\otimes c\cdot a_{\langle 1\rangle}\), \((C,\psi)\) is a coalgebra in \(\mathcal{T}(\mathcal{M}_k)^\#_A\), \((A,C,\psi)\) is a cowreath in \(\mathcal{M}_k\), \(\mathcal{M}(H)^C_A\) is just the category \(\mathcal{M}_k(\psi)^C_A\) of right entwined modules over \((A,C,\psi)\) in \(\mathcal{M}_k\). This allowed the authors to apply the results from their previous paper [Part I, Algebr. Represent. Theory 23, No. 3, 1119--1157 (2020; Zbl 1447.16031)]. In the present paper, the authors prove that: (1) If \(C\) is a Frobenius coalgebra in \(\mathcal{M}_H\) then \(F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A\) is a Frobenius functor (Proposition 2.3); (2) If \(A=H\) then \(F\colon\mathcal{M}(H)^C_H\rightarrow\mathcal{M}_H\) is a Frobenius functor if and only if \(C\) is a Frobenius coalgebra in \(\mathcal{M}_H\) (Theorem 2.4); (3) If \(C\) is a coseparable coalgebra in \(\mathcal{M}_H\) then \(F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A\) is a separable functor (Proposition 2.6); (4) If \(H\) has a bijective antipode then there is a bijective correspondence between the set of normalized Casimir morphisms for \(C\) in \(\mathcal{M}_H\) and the set of normalized Casimir morphisms for \((C,\psi)\) in \(\…
reviewed by: Małgorzata E. Hryniewicka
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 18D15
zbMATH DE Number: 6954166
zbMATH Keywords: cowreath
zbMATH Keywords: entwined module
zbMATH Keywords: quasi-Hopf algebra
zbMATH Keywords: Doi-Hopf module
zbMATH Keywords: two-sided Hopf module
zbMATH Keywords: Yetter-Drinfeld module
zbMATH Keywords: Frobenius coalgebra
zbMATH Keywords: coseparable coalgebra
zbMATH Keywords: Frobenius functor
zbMATH Keywords: separable functor
author: Daniel Bulacu
author: Stefaan Caenepeel
author: Blas Torrecillas
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.09.001
OpenAlex ID: W2890702905
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Wikidata QID: Q129326594
DOI: 10.1016/J.JALGEBRA.2018.09.001
description: scientific article; zbMATH DE number 6954166