Counting arguments for Hopf algebras of low dimension

Abstract

Let $k$ be an algebraically closed field of characteristic $0$. We show that all Hopf algebras of dimension 15, 21 or 35 over $k$ are necessarily semisimple. We also prove that Hopf algebras of dimension 25 or 49 are either semisimple or pointed. This concludes the full classification of Hopf algebras of the above mentioned dimensions. We also classify pointed Hopf algebras of dimension $pq^{2}$, where $p\neq q$ are prime numbers, and semisimple Hopf algebras of dimension 45.