Hopf bimodules, coquasibialgebras, and an exact sequence of Kac
type: Item
instance of: scholarly article
title: Hopf bimodules, coquasibialgebras, and an exact sequence of Kac
zbMATH Open document ID: 1006.16054
author: Peter Schauenburg
published in: Advances in Mathematics
publication date: 2002-07-31T00:00:00Z
review text: Recall that a matched pair of groups is a collection formed by groups \(F\) and \(G\), and actions \(\triangleright\colon G\times F\to F\), \(\triangleleft\colon G\times F\to G\) subject to certain compatibility conditions. Let \(k\) be a field and let \((F,G)\) be a matched pair of finite groups. This paper gives a theoretical explanation of two facts from the theory of Abelian extensions of Hopf algebras. The first fact is the existence of \textit{two} Hopf algebras constructed out of \((F,G)\): the group algebra \(k(F\bowtie G)\), where \(F\bowtie G\) is defined by the fact that \(F\bowtie G=F\cdot G\) is an exact factorization; and the Hopf algebra \(k^G\#kF\), the trivial extension of \(k^G\) by \(kF\) with respect to the actions induced by \(\triangleright\), \(\triangleleft\). Now, the group \(\text{Opext}(kF,k^G)\) of isomorphism classes of extensions of \(k^G\) by \(kF\) with respect \(\triangleright\), \(\triangleleft\) is related to group cohomology through the Kac exact sequence. The second fact is the existence of a map \(r\colon\text{Opext}(kF,k^G)\to H^3(F\bowtie G,k^\times)\) as part of the Kac exact sequence. The explanation is given by a very general construction, whose main idea is the following. Let \(K\hookrightarrow H\) be an inclusion of Hopf algebras with \(K\) finite dimensional. One seeks to assign to this pair another pair of the same type. Consider \(K\) as an algebra in the monoidal category \(^H{\mathcal M}\) of left \(H\)-comodules; then the category \(^H_K{\mathcal M}_K\) of \(K\)-bimodules in \(^H{\mathcal M}\) is a monoidal category with \(\otimes_K\). Assume that the inclusion is cleft. There is a tensor functor \(\omega\) from \(^H_K{\mathcal M}_K\) to \(_k{\mathcal M}\) given by \(M\to M/K^+M\). Let \(\widetilde H\) be the coalgebra of coendomorphisms of \(\omega\) (Tannakian reconstruction); this is the candidate to play the rôle of \(H\) in the new pair. However, \(\widetilde H\) is not a Hopf algebra but a coquasibialgebra; so…
reviewed by: Nicolás Andruskiewitsch
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 1747011
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Tannakian reconstruction
zbMATH Keywords: matched pairs of groups
zbMATH Keywords: Abelian extensions
zbMATH Keywords: group algebras
zbMATH Keywords: trivial extensions
zbMATH Keywords: actions
zbMATH Keywords: Kac exact sequences
zbMATH Keywords: monoidal categories
zbMATH Keywords: comodules
zbMATH Keywords: bimodules
zbMATH Keywords: coquasibialgebras
zbMATH Keywords: cleft extensions
zbMATH Keywords: double crossproducts
zbMATH Keywords: bismash products
MaRDI profile type: Publication
OpenAlex ID: W2031628781
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DOI: 10.1006/AIMA.2001.2016
description: scientific article; zbMATH DE number 1747011