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      Admissible Coalgebras and Non-cosemisimple Hopf Algebras

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          Abstract

          In this paper, we give several necessary conditions for non-cosemisimple coalgebras being admissible. The implications simplify the classification problems for Hopf algebras of dimension 45, 105 and a few others.

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          Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple

          Richard G Larson, David E Radford (1988)
          Article 0 comments Cited 59 times – based on 0 reviews     Review now
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            Finite Quantum Groups and Cartan Matrices

            Nicolás Andruskiewitsch, Hans-Jürgen Schneider (2000)
            Article 0 comments Cited 54 times – based on 0 reviews     Review now
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              Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3

              N ANDRUSKIEWITSCH, H J Schneider (1998)
              We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grad(A). Then grad(A) is a graded Hopf algebra, since the coradical A_0 of A is a Hopf subalgebra. In addition, there is a projection \pi: grad(A) \to A_0; let R be the algebra of coinvariants of \pi. Then, by a result of Radford and Majid, R is a braided Hopf algebra and grad(A) is the bosonization (or biproduct) of R and A_0: grad(A) is isomorphic to (R # A_0). The principle we propose to study A is first to study R, then to transfer the information to grad(A) via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of order p^3 (p an odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p^2; and an infinite family of pointed, non-isomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky.
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                Publication date Created: 07 September 2018
                Article
                ArXiV ID: 1809.03342
                SO-VID: 7b5c294a-7334-40e0-9462-bca292503cf4
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Comments arXiv admin note: substantial text overlap with arXiv:1509.05837
                Categories math.QA

                ScienceOpen disciplines: Algebra
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                ScienceOpen disciplines: Algebra

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