Bi-Galois objects over the Taft algebras
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 \(m-1\).
scientific article; zbMATH DE number 1436100
scientific article