Hopf bimodules, coquasibialgebras, and an exact sequence of Kac
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 that the main statement is as follows. If \(H\) is a coquasibialgebra and \(K\) is a finite dimensional Hopf subalgebra then one gets by Tannaka reconstruction a coquasibialgebra \(\widetilde H\) with \(K^*\) as a Hopf subalgebra. Several known instances are inspected in the light of this result: double crossproducts, bismash products, Radford-Majid bosonizations. In the case of extensions, a nonabelian analogue of the Kac exact sequence is investigated. In particular, if \(H=k^G\#kF\) and \(K=k^G\), then \(\widetilde H=k(F\bowtie G)\); this is the explanation to this first fact. More generally, if \(H=k^{G\tau}\#_\sigma kF\) is an arbitrary element of \(\text{Opext}(kF,k^G)\) and \(K=k^G\), then \(\widetilde H\) is the coquasibialgebra with underlying algebra and coalgebra as \(k(F\bowtie G)\) but with associativity in the category of comodules given by \(\omega\in H^3(F\bowtie G,k^\times)\) which is the image of \((\sigma,\tau)\) under the map \(r\) in the Kac exact sequence. This is the explanation of the second fact.
scientific article; zbMATH DE number 1747011
scientific article