On semisimple Hopf algebras of dimension \(2^m\).
In a previous paper [J. Algebra 232, No. 2, 617--663 (2000; Zbl 0969.16014)] the author obtained the classification of semisimple Hopf algebras of dimension 16 over an algebraically closed field \(k\) of characteristic zero. In this paper she classifies all nontrivial semisimple Hopf algebras \(H\) of dimension \(2^{n+1}\), \(n\geq 4\), such that the group \(G(H)\) of group-like elements in \(H\) is isomorphic to \(\mathbb{Z}_{2^{n-1}}\times\mathbb{Z}_2\). Amongst the class of Hopf algebras classified in this paper, there are two families dual to the examples studied by the author in [Commun. Algebra 31, No. 4, 1643-1668 (2003; Zbl 1040.16025)]. Because \(kG(H)\) is of index 2 and in view of a result of \textit{T. Kobayashi} and \textit{A. Masuoka} [Tsukuba J. Math. 21, No. 1, 55--58 (1997; Zbl 0894.16019)], a (commutative) normal Hopf subalgebra is in \(H\), and thus \(H\) is an (Abelian) extension of \(k(\mathbb{Z}_{2^{n-1}}\times\mathbb{Z}_2)\) by \(k\mathbb{Z}_2\). One of the invariants used for distinguishing isomorphism classes is the Frobenius Schur indicator of irreducible modules, as introduced by \textit{V. Linchenko} and \textit{S. Montgomery} in [Algebr. Represent. Theory 3, No. 4, 347--355 (2000; Zbl 0971.16018)].
scientific article; zbMATH DE number 2021280