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Bicrossed products with the Taft algebra

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Abstract

Let G be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra K[G] (equivalently, any bicrossed product between the aforementioned Hopf algebras) is isomorphic to a smash product between the same two Hopf algebras. The classification of these smash products is shown to be strongly linked to the problem of describing the group automorphisms of G. As an application, we completely describe by generators and relations and classify all bicrossed products between the Taft algebra and the group Hopf algebra \(K[D_{2n}]\), where \(D_{2n}\) denotes the dihedral groups.

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Authors and Affiliations

  1. “Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764 , 014700, Bucharest, Romania

    A. L. Agore & L. Năstăsescu

  2. Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    A. L. Agore

Authors
  1. A. L. Agore
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  2. L. Năstăsescu
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Correspondence to A. L. Agore.

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This work was supported by a Grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project Number PN-III-P1-1.1-TE-2016-0124, within PNCDI III. The first named author is a fellow of FWO (Fonds voor Wetenschappelijk Onderzoek - Flanders). The authors gratefully acknowledge the hospitality of Max Planck Institute für Mathematik (Bonn), where part of this work was done.

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Agore, A.L., Năstăsescu, L. Bicrossed products with the Taft algebra. Arch. Math. 113, 21–36 (2019). https://doi.org/10.1007/s00013-019-01328-3

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