The classification of triangular semisimple and cosemisimple Hopf algebras over an algebraically closed field

In this paper, we classify triangular semisimple and cosemisimple Hopf algebras over any algebraically closed field k. Namely, we construct, for each positive integer N, relatively prime to the characteristic of k if it is positive, a bijection between the set of isomorphism classes of triangular semisimple and cosemisimple Hopf algebras of dimension N over k, and the set of isomorphism classes of quadruples (G,H,V,u), where G is a group of order N, H is a subgroup of G,V is an irreducible projective representation of H over k of dimension— $\vert H\vert^{1/2}$, and u ∈ G is a central element of order less than or equal to two. This classification implies, in particular, that any triangular semisimple and cosemisimple Hopf algebra over k can be obtained from a group algebra by a twist. It also implies that (G,H,V,u) corresponds to a minimal triangular semisimple Hopf algebra over k if and only if G is generated by H and u. We then answer positively the question from our previous paper whether the group underlying a minimal triangular semisimple Hopf algebra is solvable by proving that the group H is a quotient of a central type group and hence solvable. We conclude by showing that any triangular semisimple and cosemisimple Hopf algebra over k of dimension bigger than one contains a nontrivial grouplike element. The classification uses Delignes theorem on Tannakian categories and the results of a paper of Movshev in an essential way. The proof of solvability and existence of grouplike elements relies on a theorem of Howlett and Isaacs that any group of central type is solvable, which is proved using the classification of finite simple groups. The classification in positive characteristic relies also on the lifting functor from our previous paper.