Hopf Algebras of Dimension $pq$

Abstract

Let $H$ be a non-semisimple Hopf algebra with antipode $S$ of dimension $pq$ over an algebraically closed field of characteristic 0 where $p\le q$ are odd primes. We prove that $\Tr(S^{2p})=p^2d$ where $d\equiv pq(\mathrm{mod}4)$. As a consequence, if $p,q$ are twin primes, then any Hopf algebra of dimension $pq$ is semisimple.

minor change of version 1; manuscript of the talk at the special session of AMS meeting (#986) at New York, April, 2003