Bi-Galois objects over the Taft algebras
type: Item
instance of: scholarly article
title: Bi-Galois objects over the Taft algebras
zbMATH Open document ID: 0949.16040
DOI: 10.1007/BF02810582
author: Peter Schauenburg
published in: Israel Journal of Mathematics
publication date: 2000-11-28T00:00:00Z
review text: The Taft Hopf algebra \(H_{N,m}\) is generated as an algebra by \(m\) group-like elements and a skew-primitive element. The ground field \(k\) must contain a primitive \(N\)-th root of unity. The author first shows that in \(H_N=H_{N,1}\), any \(L\)-\(H\)-bi-Galois object with \(H=H_N\) requires that \(L\cong H_N\), or equivalently, every double twist of \(H_N\) by a two-cocycle is isomorphic to \(H_N\). He computes the group \(\text{BiGal}(H_N)\) to be isomorphic to the semidirect product of \(k^*\) and \(k\). These results use \textit{A. Masuoka}'s classification of cleft extensions of \(H_N\) [Commun. Algebra 22, No. 11, 4537-4559 (1994; Zbl 0809.16046)]. These results for \(H_N\) are then extended to \(H_{N,m}\). More precisely, any \(L\)-\(H\)-bi-Galois object with \(H=H_{N,m}\) requires that \(L\approx H_{N,m}\) and when \(N\) is odd, he computes the group \(\text{BiGal}(H_{N,m})\). These results are obtained by generalizing a result of \textit{H. F. Kreimer} [Commun. Algebra 23, No. 11, 4009-4030 (1995; Zbl 0835.16034)] to give a description of the Galois objects over a tensor product of two Hopf algebras. This is effective because \(H_{N,m}\) is isomorphic to the tensor product Hopf algebra of \(H_N\) with the group-algebra of a cyclic group of order \(m-1\).
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1436100
zbMATH Keywords: Hopf-Galois extensions
zbMATH Keywords: Taft algebras
zbMATH Keywords: bi-Galois objects
zbMATH Keywords: Taft Hopf algebras
zbMATH Keywords: group-like elements
zbMATH Keywords: skew-primitive elements
zbMATH Keywords: cleft extensions
zbMATH Keywords: tensor products
reviewed by: Earl J. Taft
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1436100