On semisimple Hopf algebras of dimension \(pq^r\). (Q1878308)

Let \(k\) be an algebraically closed field of characteristic zero. This is a continuation of results relating to the author's goal of classifying certain semisimple \(k\)-Hopf algebras of dimension \(pq^r\) where \(p\) and \(q\) are distinct prime numbers, and in fact it finishes the case where \(r=2\). Let \(A\) be a semisimple \(k\)-Hopf algebra of dimension \(pq^r\), and let \(G\) be an Abelian group of order \(q^r\). Given a normalized \(2\)-cocycle \(\varphi\in(A\otimes A)^\times\) the Hopf algebra \(A_\varphi\) can be constructed, where \(A_\varphi=A\) as algebras, and the comultiplication on \( A_\varphi\) is given by \(\Delta_\varphi(a)=\varphi\Delta(a)\varphi^{-1}\), \(a\in A\), where \(\Delta\) is the comultiplication on \(A\). Suppose that \(A^*\) is of Frobenius type, that is the dimensions of the irreducible \(A^*\)-modules divide \(\dim_kA^*\), and that \(|G(A)|=|G(A^*)|=q^r\). Then there exists an invertible normalized \(2\)-cocycle \(\varphi\in kG(A)\otimes kG(A)\) such that either \(A_\varphi\) or \(A_\varphi^*\) contains a nontrivial central grouplike element. This is the main theorem of the paper. Previously, the author had obtained a classification of semisimple Hopf algebras of dimension \(pq^2\); furthermore in this case some results on the existence of nontrivial central grouplike elements were shown [\textit{S. Natale}, J. Algebra 221, No. 1, 242-278 (1999; Zbl 0942.16045); Algebr. Represent. Theory 4, No. 3, 277-291 (2001; Zbl 1013.16025)]. These results, combined with the main theorem above allow for a complete classification of semisimple Hopf algebras of dimension \(pq^2<100\) for which \(A\) and \(A^*\) are of Frobenius type. It is shown that in this case \(A\) is not simple (that is, it has no proper normal Hopf subalgebras). The classification when \(p>q\) is obtained here, which when coupled with earlier results completes the classification. Finally, the classification in the case \(pq^2<100\) is made quite explicit in certain cases, namely the until-now missing cases where \(pq^2=20,52,63\), or \(99\). It is shown that any Hopf algebra of one of these dimensions is not simple.