Hopf algebras of dimension 14

Abstract

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H_1, the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension pq where p and q are odd primes with p<q and q less than or equal to 1 + 3p, and also less than or equal to 13, is semisimple and thus a group algebra or the dual of a group algebra. We also have some partial results in the classification problem for dimension 16.