Abstract. This paper contributes to the classification problem of pq dimensional Hopf algebras H over an algebraically closed field k of characteristic 0, where p, q are odd primes. It is shown that such Hopf algebras H are semisimple for the pairs of odd

This paper contributes to the classification problems of finite dimensional Hopf algebras H over an algebraically closed field k of characteristic zero. It is shown that for a non-semisimple Hopf algebra H of dimension 18 either H or H * is pointed. Introduction.The classification of finite dimensional Hopf algebras has been developed rapidly since the end of 90's. D. Ştefan classified Hopf algebras in dimensions less than 12 [11]. It is shown by S.-H. Ng that a Hopf algebra of dimension p 2 is isomorphic to a group algebra, the dual of a group algebra or a Taft algebra [8]. For the pq dimensional Hopf algebras H where p, q are distinct primes, classification is still open in general. However many results have been obtained. N. Andruskiewitsch and S. Natale proved that 15, 21 or 35-dimensional H are semisimple [1]. Furthermore M. Beattie and S. Dȃscȃlescu settled the dimensions 14,55,65,77,91 and 143 in [2]. Other recent results of pq-dimensional cases are as follows. If primes p, q are twine primes p, p + 2 [9] or p = 2 [10] then H is semisimple by Ng. P. Etingof and S. Gelaki proved [3] that if q ≤ 2p + 1, then H is semisimple. For the case q ≥ 2p + 1, using some generalization of the method in [1] and [2], it is shown in [4] that there is no non-semisimple Hopf algebras of dimension 33

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

Using diagrammatic pictures of tensor contractions, we consider a Hopf algebra (A op ⊗ λ A * ) * twisted by an element λ ∈ A * ⊗ A op corresponding to a Hopf algebra morphism λ : A → A. We show that this Hopf algebra is quasitriangular with the universal R-matrix coming from λ when λ 2 = id A , generalizing the quantum double construction which corresponds to the case λ = id A .