Hopf Algebras of Dimension

Let H be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If H is not semisimple and dim(H) = 2n for some odd integers n, then H or H * is not unimodular. Using this result, we prove that if dim(H) = 2p for some odd primes p, then H is semisimple. This completes the classification of Hopf algebras of dimension 2p. In recent years, there has been some progress on the classification problems of finite-dimensional Hopf algebras over an algebraically closed field k of characteristic 0 (cf. [Mon98], [And02]). It is shown in [Zhu94] that Hopf algebras of dimension p, where p is a prime, are isomorphic to the group algebra k[Z p ]. In [Ng02b] and [Ng02a], the author completed the classification of Hopf algebras of dimension p 2 , which started in [AS98] and [Mas96]. They are group algebras and Taft algebras of dimension p 2 (cf. [Taf71]). However, the classification of Hopf algebras of dimension pq, where p, q are distinct prime numbers, remains open in general. It is shown in [EG98], [GW00] that semisimple Hopf algebras over k of dimension pq are trivial (i.e. isomorphic to either group algebras or the dual of group algebras). Most recently, Etingof and Gelaki proved that if p, q are odd prime such that p < q ≤ 2p + 1, then any Hopf algebra over k of dimension pq is semisimple [EG]. Meanwhile, the author proved the same result, using different method, for the case that p, q are twin primes [Ng]. In addition to that Williams settled the case of dimensions 6 and 10 in [Wil88], and Beattie and Dascalescu did dimensions 14, 65 in [BD]. Hopf algebras of dimensions 6, 10, 14 and 20 are semisimple and so they are trivial. In this paper, we prove that any Hopf algebra of dimension 2p, where p is an odd prime, over an algebraically closed field k of characteristic 0, is semisimple. By [Mas95], semisimple Hopf algebras of dimension 2p are isomorphic to