On the number of types of finite dimensional Hopf algebras
The authors construct an infinite family of pointed Hopf algebras of dimension \(p^4\) over an algebraically closed field of characteristic 0 by means of Ore extensions, where \(p\) is an odd prime number. This gives a negative answer to a conjecture of Kaplansky (1975). The same or similar counterexamples were independently and almost simultaneously found by \textit{S. Gelaki} [J. Algebra 209, No. 2, 635-657 (1998; see the next but one review Zbl 0922.16023)] and \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [J. Algebra 209, No. 2, 658-691 (1998; Zbl 0919.16027)]. The techniques in these two papers are very different from those in the paper under review. The interested reader could find the classification of all pointed Hopf algebras of dimension \(p^4\) over an algebraically closed field of characteristic 0 in [\textit{N. Andruskiewitsch} and \textit{H.-J. Schneider}, Lifting of Nichols algebras of type A2 and pointed Hopf algebras of order \(p^4\), Proc. Colloq. ``Hopf algebras and quantum groups'', Brussels 1998, Marcel Dekker (to appear)].
pointed Hopf algebras of dimension \(p^4\)
scientific article; zbMATH DE number 1295380